BIBLIOGRAPHY

The notation in the first column refers to the following rough classification into subdisciplines:

APP Applications PO Posets CL Continuous lattices POTS Partially ordered topological spaces GT General topology REF General references GLT General lattice theory TSL Topological semilattices IT Intrinsic topologies on semilattices and lattices TL Topological lattices

GLT Abbot, J.e. (1969], Sets, Lattices and Boolean Algebras, Allyn and Bacon, 1969.

TL Anderson, L.W. (1958), Topological lattices and n-cells. Duke Math. 1., vol. 25 (1958), PI', 205-208,

TL Anderson, L.W. (1959a), On the breadth and co-dimension of a topological lattice, Pac. 1. Math., vol. 9 (1959), pp. 327-333.

TL Anderson, L. W. (1959b), On the distributivity and simple connectivity of plane topological lattices, Trans. Amer. Math. Soc., vol. 91 (1959), pp. 102-112.

TL Anderson, L.W. (1959c), One dimensional topological lattices, Proc. Amer. Math. Soc., vol. 10 (1959), pp. 715-720.

TL Anderson, L.W. (1961], Locally compact topological lattices, Proc. Sympos. Pure Math. Amer. Math. Soc., vol. 2 (1961), pp. 195-197.

TL Anderson, L.W. (1%2]. The existence of continuous lattice homomorphisms, 1. London Math. Soc., vol. 37 (1962), pp. 60-62.

TL Anderson, L.W. and Ward, L.E., Jr. (1961a], A structure theorem for topological lattices, Proc. Glasgow Math. Assn .. vol. 5 (1961). pp. 1-3.

TSL Anderson, L.W. and Ward, L.E., Jr. (1961b). One-dimensional topological semilattices. 111. 1. Math (1961), pp. 182-186.

IT Atsumi, K. (1966), On complete lattices having the Hausdorff interval topology, Proc. Amer. Math. Soc., vol. 17 (1966). pp. 197-199.

GLT Aumann, G. (1955]. Bemerkung uber Galois- Verbindungen. Bayer. Akad. Wiss. Math. -Nat Kl. S. -B. (1955). pp. 281-284.

TSL Austin, e.W. (1963]. Duality theorems for some commutative semigroups, Trans. Amer. Math. Soc .. vol. 109 (1963), pp. 245-256.

APP Baartz. A. (1967], The measure algebra of a locally compact semigroup, Pac. 1. Math, vol. 21 (1967). pp. 199-214.

TL Baker, K.A. and Stralka, A.R. (1970], Compact distributive lattices of finite breadth, Pac. 1. Math., vol. 34 (1970), pp. 311-320.

GLT Balbes, R. and Dwinger, P. (1974], Distributive Lattices, Univ. Missouri Press, Columbia, 1974.

CL & GT Banaschewski, B. (1973], The filter space of a lattice, etc, Proc. Lattice Theory Conr., Houston. 1973, pp. 147-155.

CL & GT Banaschewski, B. [1977], Essential extensions of To-spaces, General Top. Appl., vol. 7 (1977), pp. 233-246.

CL Banaschewski, B. (1978a], Hulls, kernels, and continuous lattices, Houston 1. Math., vol. 4 (1978), pp. 577-525.

336 BIBLIOGRAPHY

CL Banaschewski, B. [1978b). The dualir:,' of distributive continuous lattices. preprint. 1978. Mc~aster Univ.

CL Bandelt, H.J. (1979). Complemented continuous lattices. preprint, 1979.

CL Bandelt, H.J. [1980]. The tensor product of continuous lattices. Math. Z. (to appear. 1980).

GT Birkhoff, G. (1937). Moore-Smith convergence in general topology. Annals of Math .. vol. 38 (1937), pp. 39-56.

GLT & REF Birkhoff, G. [1%7). Lattice Theory. 3rd ed., 1967. Amer. Math. Soc. Colloq. Publ., Providence. R.I.

GLT Birkhoff, G. and Frink, O. (1948), Representations of lattices by sets, Trans. Amer. Math. Soc .. vol. 64 (1948). pp. 299-316.

GLT & REF Blyth, I.S. and Janowitz, M.F. [1972], Residuation Theory, Pergamon Press. Oxford. 1972.

TSL Borrego, J.T. [1970], Continuity of the operation of a semi/attice. Coil. Math., vol. 21 (1970), pp. 49-52.

TSL Bowman, T.T. (1974), Analogue of Pontryagin character theory for topological semigroups, Proc. Amer. Math. Soc., vol. 46 (1974). pp. 95-105.

TSL Brown, n.R. [1%5]. Topological semi/attices on the 2-cell. Pac. 1. Math., vol. 15 (1965). pp. 35-46.

TSL Brown, n.R. and Stralka, A.R. (1973), Problems on compact semiiattices, Semigroup Forum. vol. 6 (1973), pp. 265-270.

CL & TSL Brown, n.R. and Stralka, A.R. [1977], Compact totally instable zero-dimensional semi/attices. Gen. Top. and Appl., vol. 7 (1977), pp.' 151-159.

GLT Bruns, G. [1961]. Distributivitiit und subdirekte Zerlegbarkeit vollstiindiger Verbiinde. Archiv d. Math. vol. 12 (1961), pp. 61-66.

GLT Bruns, G. [1961 & 1962}, Darstellungen und Erweiterungen geordneter Mengen I und II. J.f.d. reine u. angew. Mathematik, vol. 209 (1961), pp. 167-220. and vol. 210 (1962), pp. 1-23.

GLT Bruns, G. [1967]. A lemma on directed sets and chains, Arch d. Math.. vol. 18 (1967), pp. 35-43.

GLT Biichi, J. [1952}. Representation of complete lattices by sets, Port. Math. vol. 11 (1952). pp. 151-167.

GLT Bulman-Fleming, Fleischer, I. and Keimel, K. (1979). The semilattices with dislinguished endomorphisms which are equationally compact. Proc. Amer. Math. Soc .. vol. 73 (1979). pp. 7-10.

POTS Carruth, J.H. (1968). A note on partially ordered compacta. Pac. 1. Math.. vol. 24 (1968). pp. 229-231.

IT Choe, I.H. [1969a}. Intrinsic topologies in a topological lattice. Pac. 1. Math., vol. 28 (1969). pp. 49-52.

TL Cboe, T.H. [1969b]. Notes on locally compact connected topological lattices. Can. J. Math. (1969). pp. 1533-1536.

TL Cboe, I.H. [l969c]. On compact topological lattices of finite dimension, Trans. Arner. Math. Soc .. vol. 140 (1969). pp. 223-237.

TL Cboe, T.H. [1969d]. The breadth and dimension of a topological lattice. Proc. Amer. Math. Soc .. (1969). pp. 82-84.

BIBLIOGRAPHY 337

TI.. Choe, T.H. (1971], Locally compact lattices with small lattices, vol. IS (1971), pp. Sl-85.

Tl Choe, T.H. (1973a]. Injective compact distributive lattices, Proc. Amer. Math. Soc., vol. 37 (1973), pp. 241-245.

TI.. Choe, I.H. (1973b], Projective compact distributive topological lattices, Proc. Amer. Math. Soc., vol. 39 (1973). pp. 606-60S.

Tl Choe, T.H. (1969], Remarks on topological lattices. Kyungpook Math. 1., vol. 9 (1969), pp 59-62.

POTS & TL Choe, T.H. and Hong, V.H. (1976], Extensions of completely regular ordered spaces, Pac. 1. Math., vol. 64 (1976).

Tl Clark, c.E. and Eberhart, C. (1968], A characterization of compact connected planar lattices, Pac. 1. Math., vol. 24 (1968), pp. 233-240.

TI.. Clinkenbeard, D. (1976], LAttices of congruences on compact topological lattices, Dissenation, Univ. of Calif. at Riverside, 1976.

TI.. Clinkenbeard, D. (1977], Simple compact topalogical lattices, preprint, 1977.

CL & TSL Crawley, C.W. (1976], A note on epimorphisms of compact LAwson semilattices. Semigroup Forum. vol. 13 (1976), pp. 92-94.

CL & TSL Crawley, C.W. (1977], Amalgamation of compact Lawson semilattices. Shippensburg State College. P A., preprint, 1977

APP Cunningham, F. and Roy, N.M. (1974), Extreme ./imctionals on an upper semicontinuous function space. Proc. Amer. Math. Soc., vol. 42 (1974). pp. 461-465.

GI Curtis, D. and Schori, R. (1976]. rand C(X) are homeomorphic to the Hilbert cube, Bull. Amer. Math. Soc .• vol. 80 (1976). pp. 927-931.

TI.. Davies, E.B. (1968]. The existence of characters on topological Lattices. 1. London Math. Soc .• vol. 43 (1968). pp. 219-220.

APP Day, A. (1975], Filter monads, continuous lattices and closure systems, Canad. 1. Math., vol. 27 (1975), pp. 50-59.

CL Day, BJ. and Kelly, G.M. 11970], On topological quotient maps preserved by pull-backs or products. Proc. Cambridge Philos. Soc.. vol. 67 (1970), pp. 553-558.

GLT Derderian, J.e. (1967], Residuated mappings, Pac. 1. Math.. vol. 20 (1967), pp. 35-43.

GLT Dilworth, R.P. and Crawley, P. (1960), Decomposition theory for lattices without chain conditions, Trans. Amer. Math. Soc., vol. 96 (1960). pp. 1-22.

REF Dilworth, R.P. and Crawley, P. (1973), AlgebraiC Theory of LAttices, Prentice Hall, Inc., 1973.

APP Dixmier, J. (1968], Sur les espaces localement quasi-compacts, Can. 1. Math .• vol. 20 (1968), pp. 1093-1100.

OI Dowker, C.H. and Papert, D. (1966], Quotient frames and subs paces, Proc. London Math. Soc .• vol. 16 (1966), pp. 275-296.

GLT & GT Drake, D. and Thron, W.J. (1965], On the representation of an abstract lattice and the family of closed subsets of a topological space, Trans Amer. Math. Soc .• vol. 120 (1965), pp. 57-71.

GIL Dubreil'Jacotin, ML., Lesieur, L. and Croisot, R. [1953), Lefons sur La theorie des treillis, etc., Gauthier-Villars, 1953.

338 BIBLIOGRAPHY

TL Dyer, E. and Shields, A.S. [1959]. Connectivity of topological lattices. Pac. J. Math .. vol. 9 (1959). pp. 443-447.

TL Edmondson, D.E. [1956]. A non-modular compact connected topological lattice, Proc. Amer. Math. Soc .. vol. 7 (1956), p. 1157.

TL Edmondson, D.E. [1%9a]. A modular topological lattice. Pac. 1. Math.. vol. 29 (1969). pp. 271·297.

TL Edmondson, D.E. [l969b]. Modularity in topological lattices, Proc. Amer. Math. Soc .. vol. 21 (1969). pp. 81-82.

APP & CL EgIi, H. [1973]. An analysis of SCOII'S 'A-calculus models. Cornell U. Tech. Report (TR 73-191). Dec. 1973.

APP & CL EgIi, H. and Constable, R.L. [1976]. Computability concepts for programming language semantics. Theoretical Computer Science. vol. 2 (1976). pp. 133-145.

REF Eilenberg, S. and Kelly, G.M. [1966]. Closed categories. Proceedings of the. Conference on Categorical Algebra at La Jolla. 1965. Springer-Verlag. New York. 1966. pp. 421-562.

GLT Erne, M. [1979]. Order and Topology. xi+675 pp .. preprint

IT Erne, M. and Week, S. [1978]. Ordnungskonvergenz in Verbiinden. 1978.

APP & CL Ersbov, Ju. L. [1972 & 1974]. Computable functionals of finite type. Algebra i Logika, vol. 11 (1972). pp. 367-437. (Algebra and Logic. vol. 11 (1972). pp. 203-242 (1974).)

CL Ersbov, Ju. L. [1972]. Continuous lattices and A-spaces. Dokl. Akad. Nauk SSSR .. vol. 207 (1972). pp. 523-5:'6. (Soviet Math. Dokl.. vol. 13 (1972). pp. 1551-1555.)

CL Ersbov, Ju. L. [1973 & 1975]. The theory of A-spaces. Algebra i Logika. vol. 12 (1973). pp. 369-416. (Algebra and Logic. vol. 12 (1973). pp. 209-232 (1975).)

GLT Everett, c.J. [1944]. Closure operators and Galois theory in lattices, Trans. Amer. Math. Soc., vol. 55 (1944), pp. 514-525.

APP Fell. J.M.G. [1961]. The structure of algebras of operator fields. Acta Math .• vol. 106 (1961). pp. 233-280.

APP Fell, J.M.G. [1962]. A Hausdorff topology for the closed subsets of a locally compact non-Hausdorff space. Proc. Amer. Math. Soc .• vol. 13 (1%2). pp. 472-476.

IT Floyd, E.E. [1955]. Boolean algebras with pathological order topologies. Pac. 1. Math .. vol. 5 (1955), pp. 687-689.

REF Fourman, M.P., Mulvey, C.J., and Seott, D.S. [1979]. Applications of Sheaves: Proceedings, Durham, 1977. Springer-Verlag Lecture Notes in Math .• vol. 753. 1979.

TSL Friedberg, M. [1972], Metrizable approximations of semigroups, Colloq. Math., vol. 25 (1972). pp. 63-69. 164.

IT Frink, O. [1942], Topology in lauices, Trans. Amer. Math. Soc.. vol. 5 (1942). pp. 569-582.

IT Frink, O. [1954], Ideals in partially ordered sets, Amer. Math. Monthly, vol. 6 (1954). pp. 223-234.

GLT Gaskill, H.S. [1973]. Classes oj semilattices associated with an equational class oj lauices. Can. 1. Math., vol. 25 (1973). pp. 361-365.

BIBLIOGRAPHY 339

GLT Geissinger, L. and Graves, W. (1972]. The cateogry of complete algebraic lattices. 1. Combinatorial Theory. vol. 13 (1972). pp. 332-338.

CL Gierz, G. [1978]. Colimits of continuous lattices. preprint 1978. Darmstadt.

CL Gierz, G. and Hofmann, K.H. [1978], On a lattice-theoretical characterization of compact semilattices, preprint, 1978.

CL Gierz, G. und KeimeI, K. (1976], Topologische Darsrellung von Verbiinden. Math. Z., vol. 150 (1976). pp. 83-99.

APP & CL Gierz, G. und Keimel, K. [1977]. A lemma on primes appearing in algebra and anal)'sis, Houston 1. Math .. vol. 3 (1977). pp. 207-224.

CL

APP

CL & IT

TL

IT

CierI, G. and Lawson, J.D. [1979]. Generalized continuous and hypercontinuous illuices, preprint, 1979.

Giles, R. [1977]. Continuous lattices in the foundations of physics, preprint. 1977.

Gingras, A.R. [1976a], Convergence lattices, Rocky Mountain 1. Math.. vol. 6 (1976), pp. 85-104.

Gingras, A.R. (1976b]. Order convergence and order ideals, Proc. Conf Conv. Spaces, Univ. of Nevada, Reno, 1976, pp. 45-59.

Gingras, A.R. [1978], Complete distributivity and order convergence, preprint, 1978.

GLT & REF Gratzer, G. (1978], General Lattice Theory, Birkhiiuser, 1978.

GLT & REF Halmos, P. [1963], Lectures on Boolean Algebras, Van Nostrand, Princeton, 1963. (Reprinted, Springer-Verlag 1974.)

GLT Hermes, H. [1967], Ein./iihrung in die Verbandstheorie, Springer-Verlag, Berlin, 1%7.

APP Hocbster, M. [1969), Prime ideal structure in commutative rings, Trans. ArneI'. Math. Soc., vol. 142 (1969), pp. 43-60. .

CL Hoffmann, R.·E. (1979a], Continuous posets and adjoint sequences, Semigroup Forum (1979), to appear.

CL Hoffmann, R.'E. [1979b], Projective sober spaces, preprint, 1979.

CL Hoffmann, R.·E. (1979c). Sobrjflcation of partially ordered sets, Semigroup Forum, vol. 17 (1979). pp 123-138.

CL Hoffmann, R.·E. (1979d), Continuous posets and adjoint sequences, Semigroup Forum, vol. 18 (1979), pp. 173-188.

GT Hoffmann, R.-E. (197ge). Essentially complete 'I:o-spaces. Manuscripta Math., vol. 27 (1979), pp. 401-432.

CL Hoffmann, R.·E. (1980). Continuous posets, prime spectra of completely distributive complete lattices and Hausdorff compactijicatons, preprint, 47 pp ..

GT Hofmann, K.H. (1970]. A general invariant metrization theorem for compact spaces. Fund. Math, vol. 68 (1970), pp. 281-296.

CL Hofmann, K.H. [1977], Continuous lattices, topology, and topological algebra, Topology Proceedings, vol. 2 (1977), pp. 179-212.

GT & CL Hofmann, K.H. (1979). A note on Baire spaces and continuous lattices, preprint, 1979.

GLT Hofmann, K.H. and Keimel, K. (1972), A general character theory for partially ordered sets and lattices, Mem. Amer. Math. Soc., vol. 122 (1972). 121 pp.

340 BIBLIOGRAPHY

CL Hofmann, K.H. and Lawson. J.D. (1976177], Irreducibility and generation in continuous lattices. Semigroup Forum. vol. 13 (1976177). pp. 307-353.

CL Hofmann, K.H. and Lawson, J.D. (1978]. The spectral theory of distributive continuous latlices. Trans. Amer. Math. Soc.. vol. 246 (1978) pp. 285-310.

CL & TSL Hofmann, K.H. and Mislove, M. [1973]. Lawson semi/atlices do have a Pontryagin duality. Proc. Lattice Theory Conf .. C. Houston. 1973. pp. 200-205.

CL & TSL Hofmann, K.H. and Mislove, M. [1975]. Epics of compact Lawson semi/attices are surjective. Arch. Math .. vol. 26 (1975). pp. 337-345.

CL Hofmann, K.H. and Mislove, M. (1976). Amalgamation in categories with concrete duals. Algebra Universalis. vol. 6 (1976). pp. 327-347.

CL Hofmann, K.H. and Mislove, M. (1977), The lattice of kernel operators and topological algebra, Math. Z., vol. 154 (1977). pp. 175-188.

CL & TSL Hofmann, K.H., Mislove. M. and Stralka, A. (1973]. Dimension raising mt1pS in topological algebra, Math. Z., vol. 135 (1973), pp. 1-36.

CL & TSL Hofmann, K.H., Mislove, M. and Stralka, A.R. [1974]. The Pontryagin duality of compact (}-dimensional semi/auices and its applications. Springer-Verlag Lecture Notes in Math .. vol. 396, 1974.

CL & TSL Hofmann, K.H., Mislove, M. and Slralka, A.R. [1975]. On the dimensional capacity of semi/altices. Houston 1. Math .. vol. 1 (1975), pp. 43-55.

TSL & REF Hofmann, K.H. and Moslerl, P. [1966], Elements of Compact Semigroups. Merrill. Columbus. Ohio. 1966.

TSL Hofmann, K.H. and Stralka. A.R. [1973a]. Mapping cylinders and compact monoids. Math. Ann .. vol. 205 (1973). pp. 219-239.

TSL Hofmann, K.H. and Stralka, A.R. [1973b], Push-outs and strict projective limits of semilattices. Semigroup Forum, vol. 5 (1973). pp. 243-262.

CL & TSL Hofmann, KR. and Stralka, A.R. [1976]. The algebraic theory of Lawson semi/auices-Applications of Galois connections to compact semi/altices. Diss. Math .. vol. 137 (1976). pp. 1-54.

APP Hofmann, K.H. and Thayer, F.J. [198*], Almost finite dimensional C·-algebras, Diss. Math .. to appear.

GLT Horn, A. and Kimura, N. (1971], The category of semilattices, Algebra Universalis. vol. 1 (1971). pp. 26-38.

CL Hosono, Ch. and Sato, M. (1977], The retracts in R", do not form a continuous klttice, etc., Theoretical Comp. Sci., vol. 4 (1977), pp. 137-142.

IT Inse.. A.J. (1963), A compact topology for a lattice, Proc. Amer. Math Soc .• vol. 14 (1963). pp. 382-385.

IT Insel, A.J. (1964). A relationship between the complete topology and the order topology of a lattice. Proc. Amer. Math. Soc., vol. 15 (1964), pp. 847-850.

GLT & GT Isbell, J.R. (1972), Atomless parts of spaces, Math. Scand., vol. 31 (1972). pp. 5-32.

CL Isbell, J.R. (1975a). Function spaces and adjoinlS. Math. Scand .. vol. 36 (1975), pp. 317-339.

CL Isbell, J.R. (1975b). Meet-continuous klttices, Symp. Math.. vol. 16 (1975). pp. 41-54.

BIBLIOGRAPHY 341

APP Jamison, R.E. (1974). A general theory oj convexity, Dissertation, University of . Washington. 1974.

APP Jonsson, B. [1967), Algebras whose congruence lattices are distributive, Math. Scand .. vol. 9 (1967), pp. 110-121.

APP Kahn, G. (1978), Concepts Jondamentaux de theorie des modeles. preprint, 1978.

CL Kamara. M. [1978], Treillis continus et treillis comp/element dislribulij's. Semigroup Forum. vol. 16. no. 3 (1978). pp. 387-388.

GLT Keimel, K. (1972]. A unified theory of minimal prime ideals. Acta Math. Acad. Sci. Hung., vol. 23 (1972). pp. 51-69.

GT & REF Kelley, J.L. (1955], General Topology, Van Nostrand. Princeton. 1955. (Reprinted, Springer-Verlag. 1975.)

IT Kent, D.C. [1966). On the order topology in a lattice. lIlinois 1. Math.. vol. 10. (1966), pp. 90-96.

IT Kent. D.C. (1967], The interval topology and order convergence as dual convergence structures, Amer. Math. Monthly, vol. 74 (1967), pp. 426-427.

POTS Koch, R.J. [1959], Arcs in partially ordered spaces, Pac. 1. Math., vol. 9 (1959), )Jp. 723-728.

POTS Kocb. R.J. [1960) Weak cutpoint ordering in hereditarily unicoherent continua. Proc. Amer. Math. Soc .. vol. 11 (1960). pp. 679-681.

POTS Koch, R.J. [1965). Connected chains in quaSi-ordered spaces, Fund. Math .. vol. 56 (1965). pp. 245-249.

IT Kolibiar, M. [1962], Bemerkungen uber lntervalltopologie in halbgeordneten Mengen. General topology and its relations to modem analysis and algebra, Prague 1962. pp. 252-253.

POTS Krule, I.S. [1957]. Struw on the I-sphere. Duke Math. 1., vol. 24 (1957), pp. 623-626.

TIL Lan, A.Y.W. (1972a]. Concerning costability oj compact semigroups, Duke Math. 1., vol. 39 (1972), pp. 657-664.

TIL Lan, A.Y.W. [1972b]. Small semilattices. Semigroup Forum. vol. 4 (1972). pp. 150-155.

TIL Lan, A.Y.W. [1973a). Costability in SEM and TSL. Semigroup Forum. vol. 5 (1973). pp. 370-372.

TIL Lan, A.Y.W. [1973b]. Coverable semigroups. Proc. Amer. Math. Soc .. vol. 38 (1973). pp. 661-664.

TIL Lan, A.Y.W. [1975]. The boundary oj a semilaltice on an n-cell, Pac. 1. Math, vol. 56 (1975). pp. 171-174.

TIL Lan, A.Y.W. [198*]. Existence oj n-cells in Peano semilattices. to appear.

APP Larsen, K.D. and Sinclair, A.M. [1975]. Lifting matrix units in C*-algebras, II. Math. Scand.. vol. 37 (1975). pp. 167-172.

TIL Lawson, J.D. [1967]. Vietoris mappings and embeddings of topological lattices. Dissertation. Univ. of Tennessee. 1967.

TIL Lawson, J.D. [1969], Topological semi/attices with small semiJauices. 1. London Math. Soc .. vol. 2 (1%9). pp. 719-724.

342 BIBLIOGRAPHY

TL & TSL Lawson, J.D. (1970a), Lauices with no interval homomorphisms, Pac. 1. Math., vol. 32 (1970), pp. 459-465.

TSL Lawson, J.D. [1970b), The relation of breadth and codimension in topological semilattices, Duke 1. Math .. vol. 37 (1970), pp. 207-212.

TSL Lawson, J.D. [1971), The relation of breadth and codimension in topological semi/atti~es II. Duke Math. 1., vol. 38 (1971), pp. 555-559.

TSL Lawson, J.D. [1972), Dimensionally stable semi/attices, Semigroup Forum, vol. 5 (1972), pp. 181-185.

IT Lawson, J.D. (1973a), Intrinsic lattice and semi/attice topologies, Proc. Lattice Theory Conf., U. Houston. 1973, pp. 206-260.

IT Lawson, J_D, [1973b). Intrinsic topologies in topological lattices and semiiallices, Pac. 1. Math., vol. 44 (1973), pp. 593-602.

TSL Lawson, J.D. [1976a], Additional notes on continuity in semitopological semigroups, Semigroup Forum, vol. 12 (1976), pp. 265-280.

APP Lawson, J.D. [1976b], Embeddings of compact convex sets and locally compact cones, Pac. 1. Math., vol. 66, No. 2 (1976), pp. 443-453.

APP Lawson, J.D. [l976c), Applications of topological algebra to hyperspace problems, Topology-Proc. Memphis State U. Conf., Dekker, 1976. pp. 201-206.

TSL Lawson, J.D. \1977). Compact semilattices which must have a basis of subsemilattices, 1 London Math. Soc. (2), vol. 16 (1977), pp. 369-371.

CL Lawson, J.D [1979aJ. The duality of continuous posets, to appear. Houston 1. Math, to appear.

CL Lawson, J.D. [l979b), Algebraic conditions leading to continuous lattices, to appear, Proc. Amer. Math. Soc., to appear.

APP Lawson, J.D., Liukkonen, J.R. and Mislove, M.W. [1977), Measure algebras of semilatlices with finite breadth, Pac. 1. Math., vol. 69 (1977), pp. 125-139.

TSL Lawson, J.D. and Williams, W. [1970], Semi/attices and their underlying spaces, Semigroup Forum, vol. 1 (1970). pp. 209-223.

TSL Lea, J.W., Jr. (1972), An embedding theorem for compact semilattices. Proc. Amer. Math. Soc .. vol. 34 (1972), pp. 325-331

TL Lea, J.W., Jr. [1973J. The peripherality of imducible elements of a lattice, Pac. 1. Math .. vol. 45 (1973). pp. 555-560.

TL Lea, J.W., Jr. [1974aJ. The codimension of the boundary of a lattice ideal. Proc. Amer. Math. Soc .. vol. 43 (1974). pp. 36-38.

TL Lea, J.W., Jr. [1974bJ. Sublattices generated by chains in modular topological lattices. Duke Math. 1., vol. 41 (1974), pp. 241-246.

TL Lea, J.W., Jr. [1976J. Breadth two topological lattices with connected sets of imducibles. Trans. Amer. Math. Soc., vol. 219 (1976), pp. 337-345.

CL Lea, J.W., Jr. [1976177]. Continuous lattices and compact Lawson semi/attices. Semigroup Forum, vol. 13 (1976177). pp. 387-388.

TL Lea, J,W., Jr. [1978], Quasiplanar topological lauices. Houston 1. Math. vol. 4 (1978). pp. 85-90.

TSL Lea, J.W., Jr. and Lau, A.Y.W. [1975). Codimension of compact M-semilattices, Proc. Amer. Math. Soc .. vol. 52 (1975). pp. 406-408.

BIBLIOGRAPHY 343

TL Liber, S.A. [1977]. On Z·free compact lattices (in Russ.). Issled. Algebre, vyp. 5, Saratov 1977. pp. 44·52.

TL Liber, S.A. [1978], Free compact lattices, Issled. Math. Notes, vol. 24 (1978). pp. 832-835.

Cl Lystad, G. and Stralka. A. [198*]. Semi/attices having bialgebraic congruence mCliees. to appear.

REF MacLane, S. [1971]. Categories for the working mathematician, Springer-Verlag. New York. 1971. 262 pp.

APP Manna, Z. [1974]. Mathematical Theory of Computation. McGraw·Hill. 1974.

PO Markowsky, G. [1976]. Chain' complete posets and directed sets with applications. Alg. universalis. vol. 6 (1976). pp. 53·68.

CL Markowsky, G. [1977a]. A motivation and generalization of Scott's notion of a continuous lattice. preprint. .

PO Markowsky, G. [1977b]. Categories of cMin-complete posets, Theoretical Computer Science. vol. 4 (1977). pp. 125-135.

Cl Markowsky, G. [1979]. Free completely distributive lattices, Proc. Amer. Math. Soc .• vol. 74 (1979). pp. 227-234.

APP & CL Markowsky, G. and Rosen, B. [1976], Bases for cMin-complete posets, IBM 1. of Research and Development, vol. 20 (1976), pp. 138-147

IT Matsushima, Y. [1960]. Hausdorff interval topology on a partially ordered set, Proc. Amer. Math. Soc .. vol. 11 (1960). pp. 233-235.

POTS Maurice, M. [1964), Compact ordered spaces, Math. Centrum. 1964.

TSl McWaters, M. (1969]. A note on topological semi/attices, 1. London Math. Soc., vol. 1 (1969). pp. 64-6E.

GT Moore, E.H. and Smith, H.L. (1922), A general theory of limits. American Journal of Mathematics, vol. 44 (1922). pp. 102-121.

GLT l'iachbin, L. [1949), On characterizations of the lattice of all ideals of a ring, Fund. Math .. vol. 6 (1949). pp. 137-142.

POTS l'iachbin, L. [1965], Topology and Order. Van Nostrand, 1965.

IT Naito, T. [1960]. On a problem of Wolk in interval topologies. Proc. Amer. Math. Soc .. vol. 11 (1960), pp. 156-158.

APP Newman, S.E. [1969]. Me.asure algebras on idempotent semigroups, Pac. 1. Math .. vol. 31 (1969), pp. 161-169.

APP & IT Nickel, K. [1975], Verbandstheoretische Grundlagen der Intervallmathematik, Springer-Velag Lecture Notes in Computer Science. vol. 29 (1975). pp. 251-262.

TL Numakura, K. (1957], Theorems on compact totally disconnected semigroups and /.attices, Proc. Amer. Math. Soc., vol. 8 (1957), pp. 623-626.

GLT Ore, O. [1944], Galois connections. Trans. Amer. Math. Soc., vol. 55 (1944), pp. 493-513.

ClT Fapert, S. [19591. Whi::h distributive lattices are lattices of closed sets ?, Proc. Cambr. Phil. Soc .• vol. 55 (1959), pp. 172-176.

GLT Pickert, G. [1952], Bemerkungen uber Galois- Verbindungen, Archiv d. Math .. vol. 3 (1952). pp. 285-289.

344

APP&CL

APP&CL

BIBLIOGRAPHY

Plotkin, G.D. [1976), A powerdomain construction, SIAM 1. on Computing, vol. 5 (1976), pp. 452-487.

Plotkin, G.D. [1978), TWas a universal domain, 1. Compo Sys. Sci., vol. 17 (1978), pp. 209-236.

GT & GLT Priestley, H.A. [1970), Representation 0/ distributive lattices by means 0/ ordered Stone spaces, Bull. Lond. Math. Soc., vol. 2 (1970), pp. 186-190.

POTS Priestley, H.A. (1972), Ordered topological spaces and the representation 0/ distributive lattices, Proc. London Math. Soc., vol. 24 (1972), pp. 507-530.

GLT Raney, G.N. (1952), Complerely distributive complete lauices, Proc. Amer. Math. Soc., vol. 3 (1952), pp. 667-680.

GLT Raney, G.N. [1953), A subdirect-union representation lor completely distributive complete lauices, Proc. Amer. Math. Soc., vol. 4 (1953), pp. 518-522.

GLT Raney, G.N. (1960), Tight Galois connections and complete distributivity, Trans. Amer. Math. Soc., vol. 97 (1960), pp. 418-426.

IT Rennie, B.C. [1950), The Theory 0/ Lattices. Foister and Jag, 1950.

IT Rennie, B.C. [1951). Lattices. Proc. London Math. Soc., vol. 52 (1951). pp. 386-400.

TL & TSL Rhodes, J. [1973). Decomposition semilattices with applications to topalogical lattices. Pac. 1. Math .. vol. 44 (1973). pp. 299-307.

APP Roberts, J.W. [1977). A compact convex set with no extreme points. Studia Math., vol. 60 (1977), pp. 255-266.

APP Rogers, H.R. (1967). Theory 0/ Recursive Functions and Effective Computability, McGraw-Hill, 1967.

GLT Schmidt, J. [1973). Each join completion of a partially ordered set is the solution of a universal problem. 1. Aust Math. Soc .. vol. 16 (1973).

TSL Schneperman, L.B. [1968). On the theory of characters of locally bicompaci topological semigroups, Math. Sbor., vol. 77 (1968). pp. 508-532. (Math. USSR sb .. vol. 6 (1968). pp. 471-492.)

APP & CL Scott, OS. (1970), Outline 0/ a mathematical theory of computation. Proc. 4th Ann. Princeton Conference on Information Science and Systems. 1970. pp. 169-176.

CL Scott, D.S. [1972a), Continuous lattices. Springer-Verlag Lecture Notes in Math, vol. 274 (1972). pp. 97-136.

APP & CL Scott, D.s. [1972b), Lattice theory, data types, and semantics, in Formal Semantics of Programming Languages, R. Rustin Ed.. Courant Compo Sc. Symp., vol. 2 (1972). pp. 65-106.

APP & CL Scott, D.s. [1973). Models for various type-free calculi. in Logic, Methodology and Philosophy of Science IV, P. Suppes. et aL, eds., North-Holland Publ. Co .• 1973, pp. 157-187.

APP Scott, D.s. [1975a]. Combinators and classes. Springer-Verlag Lecture Notes in Computer Science. vol. 37 (1975), pp. 1-26.

APP Scott, D.S. [1975b), Some philosophical issues concerning theories of combinators, ibid pp. 346-366.

APP & CL Scott, D.S. [1976). Data types as lattices. SIAM 1. Computing, vol. 5 (1976). pp. 522-587.

BIBLIOGRAPHY 345

APP Scott, D.S. (1977). Logic and programming languages. Comm. Assoc. for Compo Mach.,vol. 20 (1977). pp. 634-641.

TL Shirley, E.D. and Stralka, A.R. [1971). Homomorphisms on connected topological lattices. Duke J. Math. (1971). pp. 483-490.

GLT Shmuely, Z. (1974). The structures of Galois connections, Pac. 1. Math .. vol. 54 (1974). pp. 209·225.

REF Sikorski, R. (1964), Boolean Algebras, Springer-Verlag, New York. 1964. (Third edition, 1969.)

APP & CL Smyth, M.B. (1977]. Effectively given domains, Theoretical Computer Science, vol. 5 (1977), pp. 257-274.

APP & CL Smyth. M.B. (1978), Power domains, 1. Comp. and Sys. Sci., vol. 16 (1978), pp. 23-36.

APP & CL Smyth. M.B. and Plotkin, G.D. (1978], The category-theoretic solution of recursive domain equations, D.A.I. Research Repon No. 60, Edinburgh, 1978.

TSL Stepp, J.W. [1971]. Semi/attices which are embeddable in a product of min intervals, Semigroup Forum, vol. 2 (1971), pp. 80-82.

TSL & TL Stepp, J.W. (1973), Topological semi/attices which are embeddable in topological lattices, 1. London Math. Soc., vol. 7 (1973), pp. 76-82.

TSL Stepp, J.W. [1975a). Algebraic maximal semi/attices, Pac. 1. Math., vol. 58 (1975), pp. 243-248.

TSL & TL Stepp, J.W. [1975b), The free compact lattice generated by a topological semi/attice, J.f.d. reine u. angew. Mathematik, vol. 273 (1975), pp. 77-86.

GL T & GT Stone, M.H. (1936), The theory of representations for Boolean algebras, Trans. Amer. Math. Soc .. vol. 40 (1936), pp. 37-111.

GL T & GT Stone, M.H. (1937), Topological representation of distributive lattices and Browerian logics. Cas. Mat. Fys .. vol. 67 (1937). pp. 1-25.

APP Stoy, J.E. (1977). Denotational Semantics - The Scott-Strachey Approach to Programming Language Theory, MIT Press. Cambridge, Mass., 1977.

Tt Stralka. A.R. (1970). Locally cOnl'ex topological lattices. Trans. Amer. Math. Soc .. vol. 151 (1970). pp. 629-640.

TL Stralka. A.R. [1971], The congruence extension property for compact topological lattices. Pac. 1. Math. vol. 38 (1971). pp. 795-802.

TSL Stralka, A.R. [1972]. The lattice of ideals of a compact semi/attice. Proc. Amer. Math. Soc .. vol. 33 (1972). pp. 175-180.

TL Stralka, A.R. [1973]. Distributive topological lattices. Proe. Lattiee Theory Conf.. U. Houston. 1973, pp 269-276.

TL Stralka. A.R. [1974]. Imbedding locally convex lattices into compact lallices. CoIl. Math .. vol. 29 (1974). pp. 144-150.

CL & TSL Stralka. A.R. [1977]. Congruence extension and amalgamation in CL, Semigroup Forum. vol. 13 (1977). pp. 355-375.

CL Stralka. A.R. (1979). Quotients of products of compact chains. Bull. Lond. Math. Soc .. vol. 11 (1979). pp. 1-4.

IT & TL Strauss, D.P. (1968). Topological lattices, Proe. London Math. Soc .. vol. 18 (1968). pp. 217-230.

GLT & GT Tbron. W.J. (1962]. Lattice equivalence of topological spaces, Duke Math. 1.. vol. 29 (1962). pp. 671-680.

346 BIBLIOGRAPHY

APP & CL Vuillemin, J. [19741, Syntaxe, Sbnantique et Axi011llltique d'un Language de Program11llltion Simple. These d'etat, University of Paris. Sept 1974.

GT Wallace, A.D. [19451. A fixed-point theorem. Bun. Amer. Math. Soc.. vol. 51 (1945). pp. 413-416.

POTS Wallace, A.D. [19541. Panial order and indecomposability. Proc. Amer. Math. Soc., vol. 5 (1954), pp. 780-781.

POTS Wallace, A.D. [1955]. Struct ideals. Proc. Amer. Math. Soc.. vol. 6 (1955). pp. 634-638.

TL Wallace, A.D. [1957a]. The center of a compact lattice is totally disconnected. Pac. J. Math. vol. 7 (1957). pp. 1237-1238.

11. Wallace, A.D. [1957b]. The peripheral character of the central elements of a lattice, Proc. Amer. Math. Soc .. vol. 8 (1957), pp. 596-597.

TL Wallace, A.D. [1957cl. Two theorems on topological lattices. Pac. 1. Math, vol. 7 (1957). pp. 1239-1241.

TL Wallace, A.D. [1958], Faetoring a lattice, Proc. Amer. Math. Soc., vol. 9 (1958). pp. 250-252.

TSL Wallace, A.D. [1961], Acyclicity of compact connected semigroups. Fund. Math .. vol. 50 (1961). pp. 99-105.

IT Ward, A.S. [1955]. On reitllions between certain intrinsic topologies in certain partially ordered sets. Proc. Cambridge Phil. Soc.. vol. 51 (1955). pp. 254-261.

APP Ward. A.S. [19691. Problem in "Topology and its Applications" (proceedings Herceg Novi 1968). Belgrade 1969, p. 352

POTS Ward, L.E., Jr. [19541. Partially ordered topological spaces. Proc. Amer. Math. Soc .• vol. 5 (1954), pp. 144-161. .

TSL Ward, LE., Jr. [1958]. Completeness in semilattices. Canad. 1. Math .• (1958). pp. 578-582.

POTS Ward, L.E., Jr. [1965al, Concerning Koch's theorem on the existence 0/ arcs, Pac. 1. Math., vol. 15 (1965). pp. 347·355.

POTS Ward, LE., Jr. [1965bl. On the conjecture 0/ R.J. Koch. Pacific J. Math., vol. 15 (1965). pp. 1429-1433.

TSL Williams, W.W. [1975), Semilauices on Peano continua. Proc. Amer. Math. Soc .. vol. 49 (1975). pp. 495-500.

GT & POTS Wilson, R.L. [1978). Intrinsic topologies on partially ordered sets and results on compact semigroups. Dissertation. Univ. of Tennessee. 1978.

GT Wojdyslawski, M. [1939). Retracts absolus et hyperspaces des continus. Fund. Math.. vol. 32 (1939). pp. 184-192.

IT Wolk, E.S. [19581. Topologies on a partially ordered set, Proc. Amer. Math. Soc.. vol. 9 (1958). pp. 524-529.

TL Wolk, E.S. [1961J. On ordel'"Convergence. Proc. Amer. Math. Soc .• vol. 12 (1961). pp. 379-384.

CL Wyler, O. [1977a]. On continuous lattices as topological algebras. preprint.

GT Wyler, O. [1977b), Injective spaces and essential extensions in TOP, General Topol. Appl.. vol. 7 (1977). pp. 247-249.

BIBLIOGRAPHY

[1] 19 Jan 19,76

[2] 19 Jan 1976

[3] 29 Jan 1976

[4] 30 Mar 1976

(S] 19 Apr 1976

[6] 28 May 1976

(7] 15 Jun 1976

(8) 28 Jun 1976

(9) 7 Jul 1976

(10] 12 Jul 1976

(11] 20 Jul 1976

(12) 1 Aug 1976

(13] 10 Aug 1976

(14) 18 Aug 1976

(15) 23 Aug 1976

(16) 1 Sep 1976

(17) 20 Sep 1976

(I8) 21 Sep 1976

[19] 30 Sep 1976

APPENDIX

Chronological List of Memos Circulated in the

Seminar on Continuity in Semilattices (SCS)

Lawson. J.D. More notes on spreads..

Hofmann. K.H. NOles on Memo [SCS-l].

Keimel. K. Equationally compact SENDOs are retracts of compact ones.

Soott.D.S. NOles on continuous lattices.

Hofmann. K.H. Notes on chains in CL-objects.

Carruth. J.H .. et aL More nOles on chains in C L-objects.

Carruth. J .H .. et aL Srill more notes on chains in CL-objects.

Hofmann. K.H. and Mislove, M.

347

On the theorem 0/ Lawson's that all compact locally connected finite dimensional semi/attices are CL

Hofmann. K.H. and Mislove. M. Commentary on Scott'sfunction spaces.

Lawson. J.D. Points with small semilarrices.

Hofmann, K.H. and Mislove, M. Errata and corrigenda to Memo [SCS-9].

Gierz, G .. Hofmann, K.H., Keimel, K. and Mislove. M. Relations with the interpolation propert)' and continuous lattices.

Keimel, K. Complements to relations with the interpolation property and continuous lattices.

Mislove.M. On Memo [SCS-I0).

Soott,D.S. Continuous lattices and universal algebra.

Hofmann, K.H. and Liukkonen, 1. The random unit interval (another example o/a CL-objecr).

Hofmann, K.H. The space of lower semicontinuous functions into a CL-object, Applications (Part I).' Copowers in CL

DaY,A. Continuous lattices and universal algebra.

Keimel, K. and Mislove, M. Several remarks.' 1. The closed subsemilattices 0/ a continuous lattice form a continuous lattice; 2. Wilen do the prime elements 0/ distributive lattice form a closed subset;

348 BIBLIOGRAPHY

3. On lower semicontinuous junction spaces; 4. On the continuity of the congruence lattice of a continuous . lattice

(20J 23 Oct 1976 Hofmann. K.H. 111 ore on the coproduct. Errata and addenda.

(21J 10 Nov 1976 Carruth, J.D., Clark, C.E., Evans, E., Lea 1.W. and Wilson. R.L. «n).

(22) 10 Nov 1976 Gierz. G. Representation of co limits in CL Part I and II.

[23J 19 Nov 1976 Lawson. 1. Non-continuous lattices.

[24J 19 Nov 1976 Hofmann. K.H., Keimel. K. Editorial

[25] 23 Nov 1976 Hofmann. K.H. Observations.

(26J 30 Nov 1976 Scott, D. A reply to an editorial

(27) 8 Dec 1976 Mislove, M. Closure operators and kernel operators in CL

[28J 15 Dec 1976 Keimel, K. and Mislove, M. The latlice of open subsets of a topological space.

(29) 28 Dec 1976 Hofmann. K.H. and Wyler, O. On the closedness of the set of primes in a continuous lattice.

(30) 4 Jan 1977 Lawson.1. Continuous semi/attices and duality.

(31) 13 Jan 1977 Hofmann. K.H. The lattice of ideals of a C·-algebra.

[32] 8 Feb 1977 Hofmann. K.H. and Lawson. J.D. The spectral theory of continuous lattices.

[33] 4 Mar 1977 Hofmann, K.H. and Lawson. J.D. Complement to Memo [SCS-32].

[3-t) 8 Apr 1977 Gierz, G. and Hofmann, K.H. On complete lattices for which O(L) is continuous-A lattice theoretical characterization of CS.

(35) 18 Apr 1977 Wyler, O. Dedekind complete posets and Scott topologies.

(36) 16 May 1977 Scott, D.S. Quotients of distributive continuous lattices; A result ofS.A. laialL

(37) 20 May 1977 Wyler, O. Comments on the spectral theory of continuous lattiCes.

(38) 1 Jul 1977 Kamara, M. Trei//is continus et treillis complements distributi/s.

(39] 15 Jul 1977 Stralka, A.R. Quotients of cubes.

(40) 28 lui 1977 Mislove, M. A new approach to some results ofGierz. Hofmann and Lawson.

(41) 25 Sep 1977 Hofmann, K.H. and Scott, D.S. An exercise on the spectrum of function spaces.

(42) 2 Nov 1977 Gierz, G. and Lawson, J.~. Generalized continuous lattices.

BIBLIOGRAPHY

(43) 18 Jan 1978

(44) 9 Feb 1978

(45) 15 Apr 1978

[46) 19 May 1978

[47) 28 May 1978

[48) 29 Nov 1978

[49) 30 Nov 1978

[50) 30 May 1979

[51) 30 May 1979

[52) 11 June 1979

349

Hofmann. K.H. Locally quasicompact sober spaces are Baire spaces.

Bauer. H. and Keimel. K. Remark on the Memo [SCS-43).

Bauer, H. Amichains and equational compacll!ess.

Gierz, G., Lawson, J.D. and Mislove, M. A result about O(X).

Hofmann, K.H. Equivalence des espaces de Batbedar er des treillis algebriques.

Hofmann. K.H. and Nino, J. Projective limits in CL and Scoer's construction.

Hofmann, K.H. and Watkins, F . .A review of a theorem of Dixmier's.

Hofmann. K.H. and Jones. L.W. Scott cominuous closure operators and modal operators. More self fUnctors to which the Scott construction applies.

Hofmann. K.H. and Watkins. F. A new Lemma on primes and a topological characterisation of the category DCL of continuous Hey/ing algebras and CL-morphisms.

Hofmann. K.H. and Keimel, K. Bemerkungen zum "Neuen Lemma':

LIST OF SYMBOLS

0* Involution of the element a in a C*-algebra 1-1.20,46

a=b Implication (in a Heyting algebra) 0-3.17,25

A « B A shields B III-US, 148

App(L) Approximating auxiliary relations on L 1-1.11,44

Aux(L) Auxiliary relations on L 1-1.9,43

~o Pederson ideal of the C·-algebra ~ 1-1.20.46

eBa Complete Boolean algebra 0-2.6,10

(CD) Complete distributivity law 1-2.4,59

eHa Complete Heyting algebra 0-2.6,10

C(L) All subsets of L closed under all infs Remarks preceding 0-3.13,23

Con(K) Compact convex subsets of K 1-1.22.50

CongA Congruences on the abstract algebra A 0-2.7(4),11

Cong- A Closer! rongruences on the compact algebra A 0-2.7(6). 12

const Constant function with value p n-4.18(2).136

COPRIMEL Cop rimes ofL 1-3.42.81

C(X,IR*) Continuous extended real-valued functions on X 0-2.10,14

d(L) The density of the lattice L III-4.16,174

D(X) Proper open lower sets of X Remarks preceding VII -3.l, 322

diag; L-tLX L The diagonal map on L 0-3.27.28 fO The corestriction of the map! 0-3.9,21

fo The inclusion induced by the map! 0-3.9.21

FiltL Filters ofL 0-13.2

FiltoL Filt l U {0} 0-13,2

FL Limit functor of F Remarks preceding 11

IV-4.l,224

Funet L Function space functor on [NFt Remarks following IV-4.9.230

g ....

The lower adjoint of the map g IV-3.4.209

(GA) Condition for a GCl-lattice III-l.17,149

G Upper graph of the function! 1-1.21,49

ifom(L,1) Continuous lattice maps ofL into I VII-2.l0,320

Hom(S,I) Continuous semilattice maps ofS into I VI-3.7,284

Hom(X,I) Continuous order-preserving maps of X into I VII-2.l0.320

I The unit interval 0-2.7(9),12 /+ Family of directed sups from / 0-4.13,34

IdL The ideals ofL 0-1.3,2

ldaL IdL U {0} 0-1.3.2

352 LIST OF SYMBOLS

ld- .A. The closed ideals of the Hausdorff ring .A. 0-2.7(7),12

(lNT) Interpolation property 1-1.15,46

lIT L The complete irreducibles of L 1-4.19,92

IRRL The irreducible elements of L 1-3.5,69

(K) Algebraic lattice axiom 1-4.4,85

K(L) The compact elements ofL 1-4.1,85 LOP The opposite lattice ofL 0-1.7.4

L The range of Pc 0-3.11.22 c Lk The range of Pk 0-111,22

lim x. The lim inf of the {x) II-l.l,98 - I

LlR' The quotient ofL by the relation R 1-2.11,61

LSC(X) LSC(X,IR*) 1-1.21.49

LSC(X,IR*) Lower semicontinuous functions from X to IR* 0-2.10.14

Ll®L2 The tensor product ofLI and L2 IV-l.44, 192

N The natural numbers 0-2.7(10), 12

N* The extended natural numbers 0-2.7(10), 13

O(i) Topology generated by all i-convergent nets II-l.l.98

O(X) The open sets in X 0-2.7(3), II

OFilt X The open filters of X 1-1.32,54

OFilto X o Filt X U {0} 1-1.32,54

°relX) The regular open sets in X 0-2.7(3), II

P The power set functor on CL Remarks following IV-4.9,230

f: FFL-+FL The map induced by P : F1.. -+ L IV-4.2(2).225

Pc The closure operator induced by the projection P 0-111,22

P(I) The random unit interval 1-2.17,64

Pk The kernel operator induced by the projection p 0-3.11.11

PL The natural map from PL to L IV-4.10(ii),231 P(IR) Distribution functions of real random variables on IR 1-2.17,64

PRIMEL The primes ofL 1-111, 70

IR The real numbers 0-2.7(3), II

IR* The extended real numbers 0-2.7(9),12 ROP The opposite relation of R 0-1.7,4

'L The sup map from Id L to L IV-4.lO(iii),231

(S) Condition fOT a Scott-open set Remarks preceding Il~ 14, 1()()

Sl The poset S with an identity adjoined 0-2.12, 15

s-< The map from L to Id L induced by -< 1-1.10,43

(SI) Strong interpolation property 1-1.15,46

(SIc) Strong interpolation property for the chain C IV-2.5(iii),195

SpecL The spectrum of the lattice L V-4.l,251

LIST OF SYMBOLS 353

Sub .A. Subalgebras of the abstract a1gebra.A. 0-2.7(5).12

Sub-JG Closed subspaces of the Hilbert space JG 0-2.7(8),12

(UF) Ultrafilter condition for a GCL-Iattice III-117. 164

USC(X.IR*) Upper semicontinuous functions from X to IR* 0-2.10,14

w(L) Weight of the continuous lattice L III-4.5, 170

w(X) Weight of the space X III-4.1, 168

WIRRL Weak irreducibles ofL V-H,246

WPRIMEL Weak primes ofL Remarks following 1-140,81

X«:y Denotes x is way-below y 1-1.1,38

.xI-II A The inverse of A under translation by x VII-l.l,306

~ Quasi-interior of X III-2.5,154

~ The sobrification of the space X V-4.9.257

ZL The natural map from Funct L to L IV-4.10, 231

f3(X} The Stone-Cech compactification of X Remark following VI-3.9,285

f(X) The closed subsets of X 0-2.7(3), 11

a The diagonal of the space 1-2.8,61

aL(a) Spec l \V L(a) V-4.l,251

v(L) The upper topology ofL U-l.17, 109

:::(L) L endowed with the lim inftopology III-3.2, 158

HL) The lim inftopology ofL III-12, 158

I(L) L endowed with the Scott topology II-2.11,118

u(L) The Scott topology ofL U-1.3.99

T(L) All lower sets closed under directed sups n-4.17,135

T(X) Closed lower sets of X Remark preceding VI -111, 286

Xu Characteristic function of the set U Il-4.18(2), 136

'l'PRIME L The pseudoprimes of L 1-123,75

o : TOP-+ POSET Functor taking a space X to X with the specialization order II-3.6,123

OX The space X with the specialization order U-16,123

(0) Condition for a set to be w-open Remark preceding III-3.20, 165

w(L) The lower topology on L III-l.l, 142

Vx Supremum of the set X 0-1.1,1

AX Infimum of the set X 0-1.1,1

lX Upper set of X 0-1.3,2

!X Lower set of X 0-1.3,2

lx Upper set of the point x 0-1.3,2

!x Lower set of the point x 0-1.3,2

354 LIST OF SYMBOLS

!x Way-below set of the point x 1-1.2,39

fx Way-above set of the point x 1-1.2,39

fM Way-above set orM II-1.28,52

ILl: I-t C Constant functor with value L Remark following IV-3.l,206

T Top or identity ofa poset 0-1.8,4

..L Bottom or zero of a poset 0-1.8,4

VL(a) Ta()SpecL V-4.l,251

cu." Topology generated by a11;-lIA for AECU VII-l.4. 306

vO Set of direct sups and filtered infs from V Remark preceding VII-l.6,307

(S-tT) Order-preserving maps from S to T 1-2.16,63

[S-tT] ~aps in (S-t T) preserving directed sups 1-2.16(iii),64

<S-tD Cofinal maps in CPosetO(S.T) IV-1.39, 191

<S) <S-t2> IV-l.41. 191 -<sup -< - sup relation on L 1-1.25,51

-,a ~egation of the element a 0-3.17,25 21' Power set of X 0-2.7(1),10

[X,Y] TOp(X.Y) with the order from Oy II-4.1,128

[XJ] Function from [X.YJ to [X.Z] induced by /: Y -t Z II-4.2.128

1\: INF-tSUPlP Functor taking an upper adjoint to its lower adjoint IV-3.3,209

vup Directed sup Remark following 1-2.3.58

[A] The order-convex hull of A VI-1.5.273

~-limit II-l.I. 98

LIST OF CATEGORIES

AL Full subcategory of CL of algebraic lattices IV-l.13. 183

ALG Full subcategory of CaNT of algebraic lattices 11-2.2,113

ArL Full subcategory of CL of arithmetic lattices IV-I.B.183

BQSOB Full subcategory of LQSOB of spaces having a basis of quasicompact open sets V-5.lS, 264

CL Full subcategory of [NFt of cor.tinuous lattices IV-1.9.182

CONT Full subcategory of UPS of continuous lattices 11-2.2,113

CPO Compact pospaces with greatest element and Remarks follOwing continuous monotone maps preserving the unit VII-3.3,323

CPoseto Continuous posets and Scott-continuous

monotone maps IV-1.39,190

CPoset Continuous posets and co final maps IV-1.39.190

CQSOB Full subcategory of BQSOB of quasicompact spaces V-5.l9.264

CS Compact semilattice monoids and continuous Remarks preceding identity preserving semi lattice morphisms V-3.26.291

CSe"'o Continuous semilattices and Scott-continuous

semilattice maps IV-1.39,190

CSem Continuous semi lattices and cofinal semi lattice maps IV-1.39, 190

DAR Full subcategory of ArLoPnHEYT of distributive arithmetic lattices with compact identity element V-5.l9.264

DCL Distributive continuous lattices with closed primes and sup-preserving lattice maps preserving way-below VIE3,323

DCPO Full subcategory of CPO of totally order-disconnected spaces VII- 3.9, p. 325

DL Continuous distributive lattices and CL-maps Remarks preceding preserving spectra V-S.16.263

DIat Distributive lattices with 0 and 1 and all 0 and 1 preserving lattice maps V-5.l9,264

GRAPH Sup-semilattices and monotone relations II-2.22,120

HEYTo Full subcategory of BEYT oflattices where the Remarks preceding

primes order-generate V-4.6,254

HEYT Complete Heyting algebras and sup-preserving Remarks preceding lattice maps 11-2.11, 118

INF Complete lattices and inf-preserving maps IV-l.l,179

INF Complete lattices and inf- and directed sup-preserving maps IV-1.9,182

LQSOB Locally quasicompact sober spaces and "perfect" maps Remarks preceding V-5.l6,263

Lat Unital lattices and lattice maps preserving the unit IV-1.l6, 185

POSET Posets and monotone maps Remarks preceding II-3.6,123

356

S SEMI SET

SOB

SUP SUp/'

TOP

UPS

~ow

LIST OF CATEGORIES

Unital semilattices and unit preserving semilattice maps

Sup-semilattices with 0 and monotone maps

Sets and functions

Sober To -spaces and continuous maps

Complete lattices and sup-preserving maps

Complete lattices and sup-preserving lattice maps

TO-spaces and continuous maps

Complete lattices and directed sup-presening maps

Up-complete posets and special. Scott-continuous

maps

Up-complete posets and Scott-continuous upper adjoints

Algebraic posets with Scott-continuous upper adjoints

IV-1.l3, 183

11-2.22, 120

Remarks preceding 11-2.4,115

Remarks preceding

V-4.7,254

11-2.2.113

Remarks following V-4.6,253

Remarks preceding

11-3.1, 121

11-2.2,113

11-1.33, 190

11-1.33,190

IV-U5.190

INDEX

Adjoint, lower see: Lower adjoint -, upper see: Upper adjoint AL-S Duality Theorem 184 IV-1.1S Alexander's Lemma, generalization of

741-3.21 Algebra, Boolean see: Boolean algebra -, Heyting see: Heyting algebra Algebra of "propositions", (of a Boolean

algebra) 13 0-2.8(5) Algebraic lattice 85 1-4.4 - and injective spaces 12611-3.15 - and Scott continuous functions 116

11-2.8, 120 11-2.22 - characterization in Scott topology

10811-1.15 -, closed order-generating subset of

244 V-2.S -, closed topologically generating subset

244 V-2.S -, completely irreducible elements of

931-4.22,244 V-2.S - has a unique smallest basis 169

III-4.4 -, irreducibles order-generate 93

1-4.23 - is a subalgebra of a power set lattice

891-4.15 - is continuous 86 1-4.5 -, Lawson topology on 147 III-I. 12 -, patch topology is compact 262

V-S.13 -, PL is algebraic 223 IV-2.28 -, products are algebraic 881-4.14 -, smallest closed order-generating sub-

set 244 V-2.5 -, subalgebra of 871-4.10 -, subalgebras are algebraic 881-4.14 -, topologically generating subset of

244 V-2.5 - when arithmetic 861-4.7 - when the way-below relation is multi-

plicative 86 1-4.7 Algebraic poset 94 1.4.28 -, open filters on 94 1-4.29, 192

IV-1.43 Algebraic semilattice 941-4.28 -, open filters on 941-4.29, 192

IV-1.43 Antitone net 2 0-1.2 Approximating extra order 194IV-2.1

Arc-chain, in a pospace 299 VI-5.S - in compact pospaces 299 VI-5.9 -, limit of in a compact pospace 299

VI-5.7 Arithmetic lattice 86 1-4.6 -, pseudoprimes in 86 1-4.8 Ascending chain condition, (for a lattice)

401-1.3(4) - and continuous lattices 421-1.7 Atom, (in a lattice) 11 0-2.7(1) Atomic lattice 11 0-2.7(1) Auxiliary order 431-1.9ff. -, approximating 441-1.11 -, interpolation property for 461-1.15 -, mUltiplicative 76 1-3.26 -, strong interpolation property for 46

1-1.15,202 IV-2.21, 202 IV-2.22 Auxiliary relation see: Auxiliary order Axiom of approximation for an auxiliary

order 41 1-1.6

Baire Category Theorem 82 1-3.43 - for continuous lattices 83 1-3.43.7 - for locally quasicompact spaces

831-3.43.8 Baire space 83 1-3.43.9 Basis for a continuous lattice see: Con­

tinuous lattice Bicontinuous lattice 317VII-2.5 see

also: Linked bicontinuous lattice Bi-Scott topology 317 Remark follow­

ing VII-2.3 -, when Hausdorff 320 VII-2.11 Boolean algebra 10 Remarks following

0-2.6, 260-3.20 -, complete 10 0-2.6 - is a continuous lattice 91 1-4.18 - is algebraic 91 1-4.18 - is arithmetic 91 1-4.18 - is atomic 91 1-4.18 - is completely distributive 91 1-4.18 -, prime element in 71 1-3.12 -, way-below relation in 391-1.3(3) Boolean lattice see: Boolean algebra Bound, lower 1 0-1.1 -, upper 1 0-1.1

C*-algebra 471-1.20 -, primitive ideal of 80 Remarks fol­

lowing 1-3.37

358

-, the Pedersen ideal of 471-1.20 Category of algebraic Heyting algebras,

and prime preserving CL-maps 189 IV-1.28

Category of algebraic lattices, and inf and directed sup preserving maps 183 IV-1.13

- and Scott continuous functions 113 11-2.2

- -- is cartesian closed 11711-2.10 - and sup and way-below preserving

maps 183 IV-1.13 Category of algebraic posets, and Scott­

continuous upper adjoints 190 IV-1.35

Category of arithmetic lattices, and inf and directed sup preserving maps 183 IV-1.13

- and sup and way-below preserving maps 183 IV-I. 13

Category of complete lattices - and complete lattice maps 179

IV-l.l - and inf and directed sup preserving

maps 182 IV-1.9 - and inf preserving maps 179 IV -1.1 - and Scott continuous functions 113

11-2.2 - - is cartesian closed 1171-2.10 - and sup preserving maps 179

IV-l.l - sup and Scott-open set preserving

maps 182 IV-1.9 Category of continuous Heyting algebras

and prime preserving CL-maps 189 IV-1.28

Category of continuous lattices, and inf and directed sup preserving maps 182 IV-1.9

- and Scott continuous functions 113 11-2.2, 13311-4.12

- - is cartesian closed 11711-2.10 - and sup and way-below preserving

maps 182 IV-1.9 - having weight less than fixed cardi­

nal 175 III-4.18 Category of continuous posets and Scott

continuous maps 190 IV-1.39, -, self duality of 191 IV-1.42 Category of continuous semilattices and

semilattice morphisms 190IV-1.39 -, self duality of 191 IV-1.42

INDEX

Category of posets and monotone maps 190 IV-1.35

Category of sup-semilattices and mOno­tone maps 120 11-2.22

Category of unital lattices and unit pre­serving lattice maps 185 IV-1.16

Category of unital semilattices and identi­ty and sup preserving maps 183 IV-1.13

Category of up-complete posets, and lower adjoints 1901V-1.33

- and Scott continuous functions, is car­tesian closed 11911-2.19

- and Scott-continuous upper adjoints 1901V-1.33

Chain 40-1.6 -, complete 80-2.1 -, gap in 94 1-4.27 - is embeddable in a cube 200

IV-2.18 -, way-below relation in 391-1.3(1) Chain Modification Lemma, for strict

chains 196IV-2.8 CL-CL op Duality Theorem 183

IV-l.l0(ii) Clopen set 140-2.9 Closed sets in a compact space, Vietoris

topology on 284 VI-3.8 - form a continuous lattice 284

VI-3.8

Closure operator 21 0-3.8 -, image is closed under directed sups

1821V-1.8 - is Scott-open 182 IV-loS -, lattice of, on a complete lattice 203

IV-2.23 - on a continuous lattice 61 1-2.9,

203 IV-2.23 - on an algebraic lattice 871-4.10, 87

1-4.9 - preserves sups 230-3.12 Closure system 23 Remarks preceding

0-3.13,230-3.13 -, closed under directed sups 24

0-3.14

Compact topology on a space 312 VII-1.16

Co-cone in a category 208 Remarks following IV-3.2

Colimit in a category 209 Remarks fol­lowing IV-3.2

INDEX

Compact convex set 801-3.39 -, closed convex subsets do not form a

continous lattice 296 Remarks fol­lowing VI-4.S

-, closed convex subsets form a conti­nuous lattice 50 1-1.22

-, converse of Krein-Milman Theo­rem 240 V-1.7

-, primes in Con(K)OP topologically ge­nerate 250 V-3.10

Compact element 381-1.1,851-4.2 - in the lattice of open sets of a

space 93 1-4.25 Compact Hausdorff space, free conti­

nuous lattice over 285 VI-3.8(ii) Compact lattice, characterization of con­

nectivity 301 VI-S.1S - has bi-Scott topology 317 VII-2.3 - has Scott and dual-Scott topology

317 VII-2.3 Compact metric semilattice, when has

small semilattices 288 VI-3.18 Compact metrizable pospace admits a ra­

dially convex metric 276 VI-1.17 Compact pospace, cocompact topology is

open lower sets 312 VII-I. 18 -, embedding in a continuous lattice

289 VI-3.22 -, free continuous lattice over 286

VI-3.lO - has a patch topology 312 VII-U8 - is PRIME L for L a distributive con-

tinuous lattic.e 323 VII-3.3 -, open upper sets are locally quasicom­

pact 312 VII-U8 -, open upper sets are order consis­

tent 312 VII-1.18

-, open upper sets form a super sober quasi-compact topology 312 VII-U8

-, totally order-disconnected 325 VII-3.8

Compact semilattice, alternate fundamen­tal theorem for 287VI-3.14

-, characterization of connectivity 300 VI-S.1l

-, characterization of continuous mor­phisms 279 VI-2.7

-, characterization of convergence in 278 VI-2.6

-, characterization of order connectivi­ty 301 VI-S.14

359

-, closed lower sets form a compact lat­tice 316VII-2.1

-, closed lower sets of 280 VI-2.lO, 286 VI-3.11

-, closed subsemilattices of 279 VI-2.8(i), 279 VI-2.9

-, compact elements of 280 VI-2.12 -, fundamental theorem for 282

VI-3.4 - has enough subinvariant pseudo­

metries 277 VI-2.3 - has patch topology of Scott topolo­

gy 313 VII-1.21 - has small semilattices at a point 287

VI-3.12 - has small subsemilattices 289

VI-3.20 - is embeddable in compact latti­

ce 316 VII-2.1 -, local minimum is compact 300

VI-S.lO -, morphisms are Lawson continu­

ous 279 VI-2.7 -, points joined by arc-chains in 300

VI-S.12 -, universal continuous lattice quotient

of 291 VI-3.26 -, way-below relation for closed lower

sets 289 VI-3.24 - when a continuous lattice 282

VI-3.4,285VI-3.9 - when a topological lattice 316

VII-2.2 - which is not a continuous lattice

296VI-4.5 Compact semi topological semilattice, has

closed graph 332 VII-4.8 - is topological 332 VII-4.8, 333

VII-4.11 Compact totally disconnected semilattice,

fundamental theorem for 287 VI-3.13

Compatible topology, for a poset 272 VI-1.2

Complemented lattice 10 0-2.6 Complete Boolean algebra see: Boolean

algebra Complete chain 8 0-2.1 - is a continuous lattice 421-1.7 -, when algebraic 94 1-4.27 Complete Heyting algebra see: Heyting

algebra Complete lattice 80-2ff.

360

-, and Scott continuous functions 113 11-2.2, 11511-2.4, 11511-2.5

-, characterization when linked bicon­tinuous 318 VII-2.8

-, completely prime filter 257 V -4.10 -, definition of the spectrum 251

V-4.1 -, function space is a Heyting algebra

13511-4.17 -, function space is a continuous lat­

tice 12911-4.7 -, function space is meet-continu­

ous 13511-4.17 - has continuous Scott topology 329

VII-4.4 - has join-continuous Scott topology

329 VII-4.4 -,hull of an element 251 V-4.l -, hull-kernel topology on the spec-

trum 252 V -4.3 -, interval topology on 318 VII-2.7 -, irreducible elements order-generate

245V-2.7 - is a complete Heyting algebra 71

1-3.13 - is completely distributive 721-3.15 - is meet-continuous and distributive

711-3.13 - is topological in the lower topology

144 III-l.4 -, lattice of congruences on 203

IV-2.25 -, lattice of continuous kernel opera­

tors 203IV-2.25 -, Lawson closed subsemilattice of

163 III-3.14 -, Lawson topology is productive 153

III-2.2 -, Lawson topology on 144

111-1.5 see also: Lawson topology -, lower topology on 142 III-1.1 see

also: Lower topology -, order-generating subset of 701-3.8,

245 V-2.7 -, patch topology on primes 261

V-5.l2 -, patch topology on primes is func­

torial 324 VII-3.6 -, quasicompact subsets of spectrum

258 V-5.l -, saturated quasicompact set in spec­

trum 258 V-5.3

INDEX

-, saturated subset of the spectrum 258 V-5.2

-, Scott topology is a continuous lat­tice 13311-4.11, 13411-4.14, 135 11-4.17,13911-4.26

-, Scott topology is sober 134 11-4.14 -, smallest closed order-generating set

in 245 V-2.7 -, sober subspaces in the lower topolo­

gy 256 V-4.8 -, spectrum is sober 252 V-4.4 -, strictly embedded set in spectrum

260 V-5.9 -, tensor product on 192IV-1.44 -, when a topology contains Scott

topology 309 VII-1.9 -, when a topology is Scott topology

309 VII-1.9 -, when bi-Scott topology is Hausdorff

320 VII-2.11 -, when compact pospace 328 VII-4.2 -, when interval topology is Hausdorff

318 VII-2.8 -, when Scott topology is continu­

ous 327 VII-4.1, 328 VII-4.2 -, when Scott topology is produc­

tive 313 VII-1.20 -, when Scott topology is sober locally

quasicompact 327 VII-4.1 -, when spectrum of Scott topology

327VII-4.1 Complete semilattice 150-2.11 -, closed lower sets of 279 VI-2.8(ii) -, Iim-inf convergence in 10811-1.16 -, order consistent topology on 109

11-1.17 -, Scott continuous function on 120

11-2.21 -, Scott topology on 10811-1.16 -, when continuous 65 1-2.20 Complete-continuous semilattice 52

1-1.29 - and injective spaces 12711-3.19 -, closure properties of 651-2.21 - is embeddable in a continuous lat-

tice 10911-1.18(iii) -, Scott topology on 10911-1.18 Completely distributive algebraic lat­

tice 333 VII-4.1O

Completely distributive lattice 59 1-2.4, 591-2.5

- and injective spaces 12611-3.13

INDEX

- and Scott continuous functions 120 11-2.20

-, coprimes in 240 V-1.9 -, coprimes form a continuous poset

241 V-l.l0 - is continuous 721-3.15 - is embeddable in a cube 204 IV-

2.30 - is embedded in a product of intervals,

Remarks following 72 1-3.15 - is linked bicontinuous 318 VII-2.9 -, when hypercontinuous 205IV-2.31 Completely irreducible element 91

1-4.19, 91 1-4.20 Cone, over a diagram 206IV-3.1 Congruence relation 11 0-2.7(4) -, on a continuous lattice 621-2.12 Construction of F-algebras, by a pro-con-

tinuous functor F 230IV-4.9 Construction of function space algebras,

on INFt 232IV-4.12 Construction of Scott topology algebras,

on JNFt 233IV-4.1S Construction of ideal algebras, on INFt

232IV-4.14 Construction of power set algebras, on

CL 232IV-4.13 Continuous Heyting algebra, and prime

preserving maps 189IV-1.27 -, dual to sober locally quasicompact

spaces 263 V -5.16, 2'66 V -5.25 -, function space on 264 V-S.20 -, is a distributive continuous lattice

263 V-S.16 -, tensor productfor 265 V-S.21 Continuous lattice 41 1-1.6 -, and Scott continuous functions 113

11-2.2,11611-2.8 -, base for the Scott topology of 104

11-1.10 -, basis for 168 III-4.1 -, characterization in Scott topology

10711-1.14, 10411-1.8 -, CL-congruence on 204IV-2.27 -, closed order-generating subset 243

V-2.1 -, closure operator on 61 1-2.9 -, closure properties of 601-2.7 -, congruence on 621-2.12 -, Dedekind cuts in 175 III-4.19, 202

IV-2.22 -, density of 174 III-4.16

-, distributive see: Continuous Heyting algebra

361

-, equational characterization of 58 1-2.3

-, free 901-4.17 -, free over a compact pospace 286

VI-3.10 -, free over a compact space 285

VI-3.8(ii) -, free over a set, Remarks follow­

ing 285 VI-3.9 -, function space is a continuous lat-

tice 12911-4.7 - has a minimal basis 169 III-4.4 - has countable basis 172 III-4.10 - has small semilattices 282 VI-3.4 -, homomorphism of 601-2.6 -, ideal lattice of 245 V-2.6 -, lNFt -maps preserve irreducibles

245 V-2.8 -, interval topology is Hausdorff 166

III-3.23 -, irreducible elements in 691-3.7,

243 V-2.1, 244 V-2.4 -, irreducible in the function space of

13511-4.18, 13611-4.19 - is a compact semilattice 155

III-2.13, 204 IV-2.28 - is a Heyting algebra 71 1-3.14 - is a quotient of an arithmetic lattice

891-4.16 - is completely <iistributive - is embeddable in a cube

IV-2.19

721-3.15 201

- is embeddable in a product of chains 200IV-2.17

- is hypercontinuous 166 III-3.22 - is injective in the Scott topolo-

gy 123 11-3.5 - is meet-continuous 441-1.12,57

1-2.2 - is quotient of arithmetic of same

weight 171 III-4.8 -, kernel operator on 61 1-2.11,62

1-2.13, 61 1-2.9 -, lattice of CL-congruences on 204

IV-2.27 -, lattice of continuous kernel opera­

tors 203 IV-2.24, 204 IV-2.26 -, Lawson closed subsemilattice

of 146 III-l.l1 -, Lawson topology 144 III-LS

362

-, Lawson topology is compact 146 III-1.10

-, Lawson topology is compact met­ric 172 III-4.10

-, Lawson topology is the interval topo­logy 166 III-3.23

-, order-generating sets in 244 V-2.4 -, order-generating sets in Id L 245

V-2.6 -, order-generating subset of 244

V-2.4 -, patch topology on primes is Haus­

dorff 324 VII-3.5 -, PL is the lower topology 223

IV-3.28 -, primes order-generate 71 1-3.14 -, primes topologically generate 250

V-3.9 -, projection operator on 631-2.14,

631-2.15 -, properties of weights on 173 III-

4.14 -, pseudoprimes 751-3.23,246Re­

marks following V-3.1 -, pseudoprimes are weak primes 248

V-3.5 -, quotient of 621-2.12 -, relation of weight and density 174

I11-4.17 -, Scott topology h~s basis of open fil­

ters 10711-1.14 -, Scott topology has same weight 171

III-4.9 -, Scott topology is Baire 10611-1.13 -, Scott topology is locally quasicompact

sober 106 11-1.13 -, Scott topology is super sober 310

VII-1.12 -, Scott topology is the upper topolo­

gy 166 III-3.23 -, Scott-open sets in 10011-1.5(5) -, smallest approximating relation on

451-1.14 -, smallest closed order-generating set

243 V-2.1 -, smallest closed topologically gener-

ating set 244 V-2.4 -, subalgebra of 60 1-2.6 -, tensor prc,ulCt on 192 IV-1.45 -, topologic~ 111) generating sets in Id L

245V-2.6 -, topologically generating subset 243

V-2.3, 244 V-2.4 -, way-below relation on function

spaces 13711-4.20

INDEX

-, weak irreducibles 246 V-3.1 -, weak irreducibles are closed 246

V-3.2 -, weak primes 246 V-3.1, 247 V-3.4 -, weak primes are closed 246 V-3.2 -, weak primes are weak irreducibles

248 V-3.6 -, weak primes equal weak irreducibles

250V-3.11 -, weight is multiplicative 173 III-4.14 -, weight of 170 III-4.5, 221 IV-3.24 -, weight of projective limit 222

IV-3.25 Continuous poset 521-1.26, 631-2.16,

641-2.17, 190IV-1.39_ -, and sober spaces 12711-3.18 -, function space on 13811-4.21 -, open filterin 781-3.31, 191

IV-1.40 -, Scott topology for 10911-1.19 -, Scott topology has basis of open fil-

ters 11011-1.21 -, Scott topology is locally quasicom-

pact 10911-1.20 -, Scott topology is sober 10911-1.20 -, way-below relation in 521-1.27 Continuous semilattice 52 1-1.29, 64

1-2.18,190 IV-1.39 -, irreducibles order-generate 79

1-3.34 - is"meet-continuous 651-2.19 -, open filters on 191 IV-1.40 Converse relation 4 0-1.7 Co-prime element 701-3.11,81

1-3.42 -, form an up-complete poset 811-

3.42

Dense element, in a lattice 83 1-3.43.5 Density see: Continuous lattice Diagram, cone over 206 IV-3.1 -, in a category 206 IV-3.1 Direct limit, in a category 209IV-3.2 Directed net 20-1.2 Directed set 2 0-1.1 Discrete category 207 Remarks follow­

ing IV-3.1 Distributive algebraic lattice, compact­

open subsets are base for the spec­trum 263 V-5.18

-, duality involving sober spaces 263 V-5.18

-, hull of a compact element 263 V-5.18

INDEX

-, primes are closed 249 V-3.7 Distributive arithmetic lattice, dual to dis­

tributive lattices 264 V-S.19 -, duality involving quasicompact sober

spaces 264 V-S.19 -, primes are closed 249 V-3.7 -, spectrum totally order-disconnec-

ted 325 VII-3.8 Distributive arithmetic lattices, Priestly

duality for 325 VII-3.9 Distributive complete lattice, is bicontin-

uous iff linked bicontinuous 318 VII-2.9

-, saturated quasicompact set in spec­trum 259 V-S.4

Distributive continuous lattice, is Scott

topology of spectrum 259 V-S.S - is the lattice of open lower sets if the

spectrum is closed 324 VII-3.7 - is topological lattice 317 VII-2.4 -, open filter in 71 1-3.12 -, patch topology is Lawson topology

261 V-S.12 -, patch topology is compact 261

V-S.13 -, prime element in 71 1-3.12 -, primes are closed 249 V-3.7 -, pseudoprime in 751-3.24 -, pseudoprimes equal weak primes

248 V-3.S -, spectrum is sober locally quasicom­

pact 259 V-S.5 -, when way-below is multiplicative

249 V-3.7. 261 V-S.13 Distributive continuous semilattice,

primes order-generate 791-3.35 Distributive lattice 10 0-2.6 ' -, ideal lattice is distributive arithmetic

325 VII-3.8 -, prime element in 701-3.12 -, prime ideal in 731-3.19, 751-3.22 -, way-below relation in 75 1-3.22 Distributive semilattice 76 1-3.29 Duality of lI:lgebraic lattices and com-

pletely distributive algebraic lat­tices 333 VII-4.1O

Duality of compact pospaces and distribu­tive continuous lattices 324 VII-3.4

Duality of compact semilattices and dis­tributive continuous lattices 331 VII-4.6

363

Duality of continuous lattices and com­pletely distributive lattices 333 VII-4.1O

Duality of continuous posets and com­pletely distributive lattices 241 V-UO

Duality of distributive arithmetic lattices and totally order-disconnected pos­paces 325 VII-3.8

Element, compact see: Compact ele­ment

Equalizer, in a category 207 Remarks following IV-3.1

Evaluation map, on [L~A] 234 IV-4.17

Extra order, on a complete lattice 194 IV-2.1

Extremally disconnected space 140-2.9

Family of finite character, in a power set 10011-1.5(4)

Filter 2 0-1.3 -, open 69 1-3.3 -, prime 731-3.17 -, principal 20-1.3 -, Scott-open 99 III-1.3 Filtered inf preserving function 4 0-1.9 Filtered net 2 0-1.2 Filtered set 2 0.1.1 Fixed point theorem 80-2.3 Free continuous lattice 901-4.17, see

also: Continuous lattice Function, arbitrary inf preserving 4 0-1.9

-, arbitrary sup preserving 4 0-1.9 -, directed sup preserving 40-1.9 -, filtered inf preserving 4 0-1.9 -, finite inf preserving 40-1.9 -, finite sup preserving 40-1.9 -, idempotent 90-2.5, 20 Remark

preceding 0-3.6 -, lower semicontinuous 140-2.10 -, monotone 40-1.9 -, order preserving 4 0-1.9 -, semicontinuous 140-2.10 -, upper semicontinuous 140-2.10 Function space functor Funct on

INFi 218IV-3.18 -, preserves injective (surjective) maps

219IV-3.19 -, preserves projective limits 219

IV-3.19

364

Galois adjunction see: Galois connec­tion

Galois connection 180-3.1, 18 0-3ff. GeL-lattice see: Generalized continu­

ous lattice Generalized Baire Category Theorem

821-3.43 Generalized continuous lattice 149

III-1.17 -, closure properties of 165 III-3.18 -, iff Scott topology hypercontinu-

ous 333 VII-4.12 -, Lawson topology is Hausdorff 164

III-3.1S -, Scott cluster points of ultrafil­

ters 164 III-3.17 -, sup-map characterization of 165

III-3.19 Greatest lower bound 2 0-1.1

Hausdorff space is locally compact, iff O(X) is continuous 260 V-S.7

Heyting algebra 25 0-3.17, 71 1-3.13 -, and the Scott topology 13411-4.1S,

13411-4.16 -, as a function space 13511-4.17 -, closure properties of 27 0-3.26 -, complete 10 0-2.6 -, dual to sober spaces 266 V-S.24 -, homomorphism of 270-3.24 - is meet continuous 31 0-4.3 -, subalgebra of 270-3.24 -, when a continuous lattice 71 1-3.14 Hilbert space 120-2.7(8) Homomorphism 5 Remarks following

0-1.9 -, of continuous lattices 601-2.6 -, of Heyting algebras 270-3.24 Hypercontinuous lattice 166111-3.22 - is continuous 166111-3.22 -, Scott topology is the upper topolo-

gy 166 III-3.23 -, way below relation in 166 III-3.22

Ideal 2 0-1.3 -,prime 731-3.17 -, principal 2 0-1.3 Ideal functor, on INFt 221 IV-3.23 -, preserves injective (surjective) maps

221IV-3.23 -, preserves projective limits 221

IV-3.23

INDEX

Idempotent function 9 0-2.5, 20 Remark preceding 0-3.6

Identity, (of a semilattice) 40-1.8 Implication, (in a Heyting algebra) 25

0-3.17 lNF-SUPDuality Theorem 179IV-1.3 Infimum 2 0-1.1 Injective space 121 11-3.1 - and algebraic lattices 12611-3.15 - and complete-continuous semilat-

tices 12711-3.19 -, characterization of 122 11-3.4 -, closure properties of 121 11-3.2 -, equivalent conditions for 12611-

3.13 -, function space on 12911-4.6 -, function space is a continuous lat-

tice 129 11-4.6 - is a continuous lattice 12411-3.7 Interpolation property, see: Auxiliary or-

der Interval topology 151 III-1.22 - is Hausdorff 166 III-3.23 - is the Lawson topology 166 III-3.23 Irreducible closed set 79 1-3.36, 106

Remark preceding 11-1.12 Irreducible element 691-3.4, 69 1-3.S -, in a continuous lattice 691-3.7 -, order-generate a continuous lattice

701-3.10 Isomorphism 4 0-1. 9

Join-continuous lattice 30 0-4.1 Join-irreducible element 69 Remarks

following 1-3.4

Kernel operator 21 0-3.8 - has continuous image 181 IV-1.7 -, lattice of continuous ones is alge-

braic 203IV-2.24 -, lattice of continuous ones is continu­

ous 203IV-2.24 -, lattice of, on a complete lattice 203

IV-2.23 -, on a continuous lattice 61 1-2.10,

203IV-2.24 -, on an algebraic lattice 203IV-2.24 -, preserves directed sups 181 IV-1.7 -, preserves infs 230-3.12 Koch's Arc Theorem 299 VI-S.9

INDEX

Lat-ArLOP Duality Theorem 185 IV-1.17

Lattice 4 0-1.8 -, algebraic see: Algebraic lattice -, arithmetic see: Arithmetic lattice -, complemented 10 0-2.6 -, complete 80-2.1 see also: Com-

plete lattice -, completely distributive see: Com-

pletely distributive lattice -, continuous see: Continuous lattice -, distributive 10 0-2.6 -, join-continuous see: Join-continu-

ous lattice -, M-distributive 651-2.22 -, meet-continuous see: Meet-continu-

ous lattice -, modular see: Modular lattice -, prime element in 701-3.11 -, prime filter in 731-3.17 -, prime ideal in 731-3.16, 731-3.17 -, Scott topology on 9911-1.3 see al-

so: Scott topology -, topological 274 VI-1.11 Lattice of closed congruences, of a topo­

logical algebra 12.0-2.76 Lattice of closed ideals, of a C*-alge­

bra 471-1.20 -, primes in 80 1-3.37 Lattice of closed ideals of a ring 12

0-2.77 Lattice of closed subsets of a space 11

0-2.73 Lattice of closed subspaces, of a Hilbert

space 12 0-2.78 Lattice of compact normal subgroups, is

algebraic for almost connected groups 941-4.26

Lattice of congruence relations, on an alg~bra 110-2.74

Lattice of filters, of a semilattice 13 0-2.82

Lattice of ideals, of a lattice 12 0-2.74(iii), 13 0-2.8(3)

Lattice of lower sets, of a poset 13 0-2.8(1)

Lattice of monotone functions, on the unit interval 120-2.7(9)

Lattice of normal subgroups, of a group 110-2.7(4i)

Lattice of open sets 11 0-2.7(3) - is a continuous lattice 421-1.7,42

1-1.8

365

- is algebraic 93 1-4.25 - is arithmetic 94 1-4.25 -, way-below relation in 401-1.4 Lattice of partial functions, on the natural

numbers 12 0-2.7(10) -, from X to Y 531-1.31 Lattice of regular open sets 27 0-3.22 Lattice of Scott-open sets, co-prime in

10511-1.11 - is a continuous lattice 10711-1.14 - is completely distributive 107

11-1.14 -, on a continuous lattice 10511-1.11 -, prime in 10511-1.11 Lattice of subalgebras, of an algebra 12

0-2.7(5) Lattice of theories, of a Boolean alge­

bra 13 0-2.8(5) Lattice of two-sided ideals, of a ring 11

0-2.74ii - is a continuous lattice 421-1.7 -, way-below relation in 401-1.35 Lattice of upper sets, of a poset 13

0-2.8(1) Lawson topology 144 III-1.5 -, closed lower set in 144 I1I-1.6 -, continuous function for 145 III-1.8 - has small compact semilattices 156

III-2.13 - has small open semilattices 155

III-2.11, 155 III-2.13 - has small open-closed semi lattices

156 III-2.16 - is coarser than the lim-inf topolo-

gy 159 III-3.5 - is compact Hausdorff 155 III-2.13 - is compact metric 172 III-4.10 - is compact zero-dimensional 156

III-2.16 - is Hausdorff 164 III-3.15, 161

III-3.9 - is productive 153 III-2.2 - is quasicompact 150 III -1.21 (iii),

146 III-1.9 - is the interval topology 166 III-3.23 - is the lim-inf topology 161 III-3.9 -, on a continuous lattice see: Con-

tinuous lattice -, on an up-complete poset see: Up-

complete poset -, open lower sets in 165 III-3.20 -, open upper set in 144 III-1.6 Least upper bound 1 0-1.1

366

Lim-inf convergence, is Scott conver­gence 10411-1.8

Lim-inf topology 158 1II-3.2 -, agrees with the Lawson topology

1591II-3.6 -, closed lower set in 159 1II-3.3 -, closed set in 159 1II-3,4 - is Hausdorff 160 1II-3.7, 160

III-3.8 - is finer than the Lawson topology

159 III-3.5 - is quasicompact 159 II1-3.6 -, on a continuous lattice 162111-3.11 -, open upper set in 159 III-3.3 Limit, of a diagram 207 IV-3.1 Limit cone, over a diagram 207 IV-3.1 Limit maps 207IV-3.1 Limit natural transformation 207

IV-3.1 Limit preserving functor, on a category

214IV-3.1O

Linked bicontinuous lattice 317 VII-2.5 - is a compact lattice in Lawson topolo­

gy 317 VII-2.6 - is completely distributive 318

VII-2.9 - is embeddable in a cube 318 VII-

2.9 -, Lawson topology has small lattices

317 VII-2.6 Locally quasicompact space 40I-I,4(ii) -, co-compact topology on 267 V-5.26 -, lower topology on the closed

sets 1471II-1.13 -, open sets form a continuous lattice

421-1.7 -, patch topology is Hausdorff 324

VII-3.5 -, Scott topology on the lattice of closed

sets 147I1I-1.13 -, tensor product of topologies 265

V-5.21 -, way-below relation on the closed

sets 147 II1-1.13 Locally quasicompact spaces, duality with

continuous Heyting algebras 263 V-5.16

Lower adjoint 180-3.1 - is a lattice map 188IV-1.26 - is injective 200-3.7 - is Scott continuous 11411-2.3(1) - is surjective 200-3.7

INDEX

- preserves compact elements 183 IV-1.11

- preserves Scott-open sets 180 IV-l,4

- preserves the way-below relation 183 IV-1.11, 180 IV-l,4

Lower bound 1 0-1.1 Lower limit of a net 9811-1.1 Lower semicontinuous function 14

0-2.10, 9B-Remarks preceding 11-1.1 -, form a continuous lattice 491-1.21 - is Scott continuous 114 11-2.3(3) Lower set 20-1.3 Lower topology 142 III-Ll -, continuous function for 143111-1.2 -, forms a continuous lattice 148

1II-1.16 -, is productive 143 III-1.3 -, open subset of 165 III-3.20 -, way-below relation on 148111-1.14

Meet-continuous lattice 300-4ff. -, auxiliary orders on 441-1.12 -, closed lower sets in 153 III-2.1 -, closure properties of 33 0-4.10 - has continuous Scott topology 329

VII-4.4 - is a continuous lattice 451-1.14,66

1-2.23, 155 III-2.13, 156 III-2.14, 154 III-2.9, 163 II1-3.13

- is a generalized continuous lattice 156 III-2.14

- is an algebraic lattice 156 II1-2.16 - is compact pospace 329 VII-4.4 - is super sober 310 VII-1.12 - is topological in the Scott topolo-

gy 310 VII-1.13 -, Lawson topology has small semilat­

tices 155 I1I-2.ll, 155 III-2.13 -, Lawson topology is compact Haus­

dorff 155 I1I-2.13 -, Lawson topology is Hausdorff 153

III-2.4 -, Lawson topology is semitopologi­

cal 153 III-2.3 -, Lawson topology is zero-dimension­

al 156 III-2.16 -, open filters are a base for the Scott

topology 155 III-2.11 -, open upper sets in 153 II1-2.1 -, Scott topology 13411-4.16 -, Scott topology is a continuous lattice

155 III-2.13

INDEX

-, Scott topology is a dual Heyting alge­bra 13411-4.15

-, Scott topology is a Heyting algebra 13411-4.16

-, Scott topology is join-continuous 134 11-4.15, 134 11-4.16

-, way-below relation in 41 1-1.5,45 1-1.13

-, when the Scott topology is productive 310 VII-1.13

Meet-continuous semilattice 33 0-4.6 Meet-irreducible element see: Irreduc­

ible element Modular lattice, irreducible element in

77 1-3.28 Monotone convergence space 125

11-3.9 -, continuous function on 12811-4.2 -, function space is a continuous lattice

128 11-4.3 -, function space on 128 11-4.2, 128

11-4.3 Monotone function 4 0-1. 9 Monotone net 2 0-1.2 Monotone normal pospace is embeddable

in a cube 275 VI-LIS Multiplicative function 5 Remarks fol­

lowing 0-1.9

Negation, (in a Heyting algebra) 25 0-3.17

Net 20-1.2 -, antitone 20-1.2 -, directed 20-1.2 -, filtered 20-1.2 -, lower limit of 9811-1.1 -, monotone 2 0-1.2

O-regular topology, for a poset 307 VII-1.6

Open filter 68 Remarks following 1-3.2 -, in a continuous lattice 69 1-3.3 Open upper set, in a complete lattice 68

1-3.1, 68 1-3.2 -, maximal elements in the comple-

ment 69 1-3.4 Operator, closure see: Closure operator -, kernel, see: Kernel operator -, projection, see: Projection operator Opposite relation 4 0-1.7 Order, auxiliary see: Auxiliary order

367

Order consistent topology 108 11-1.16 Order preserving function 4 0-1. 9 Order regular topology, for a poset 307

VII-I. 6 Order topology 151 III-1.23 Order-generating set, in a complete lat­

tice 701-3.8, 701-3.9

Partial function 12 0-2.7(10) Partial order, closed 272 VI-1.1 -, lower semicontinuous 271 VI-1.1 -, semicontinuous 271 VI-1.1 -, upper semicontinuous 271 VI-1.1 Partially ordered set see: Poset Patch topology 261 V-s.ll - is functorial 324 VII-3.6 - is Hausdorff on a locally quasicom-

pact space 324 VII-3.s -, on a space 261 V-S.ll, 312

VII-l.16 -, on an algebraic lattice 262 V-S.13 -, on the primes is compact 261

V-S.13 -, on the primes is the Lawson topolo­

gy 261 V-S.12 Pedersen ideal, of a C*-algebra 47

1-1.20

Poset 4 0-1.6 -, algebraic, see: Algebraic poset -, cocompact topology is order consis-

tent 312 VII-1.17 -, continuous, see: Continuous poset - has an order consistent topolo-

gy 307 VII-1.7 -, ideals form an algebraic poset 95

1-4.30 - is compact pospace if semicontinu­

ous 311 VII-l.1S -, o-regular topology 307 VII-1.6 -, order convex hull of a subset 273

VI-1.s

-, order convex subset 273 VI-1.s -, order regular topology for 307

VII-1.6 -, saturated quasicompact sets are Scott

closed 312 VII-1.17 -, Scott topology on 139 11-4.25 -, up-complete 150-2.11 -, w-point in 307 VII-1.6 -, when the patch topology is com-

pact 313 VII-1.l9

368

-, when a pospace in the patch topolo­gy 312VII-1.17

-, when upper sets are saturated 312 VII-1.17

-, with compatible topology 272 VI-1.2

Pospace, arc-chain in 299 VI-5.5 - is locally convex if compact 273

VI-1.9 - is monotone normal if compact 273

VI-1.8 -, local minimum in 299 VI-5.8 -, lower semicontinuous 271 VI-I. 1 -, monotone normal 273 VI-1.7 -, open upper sets form an o-regular

topology 307 VII-1.7 -, open upper sets form an order consis­

tent topology 307 VII-1.7 -, radially convex metric for 275

VI-1.16 -, semicontinuous 271 VI~1.1 -, upper semicontinuous 271 VI-1.1 -, when each point is sup of OJ-points

307 VII-1.7 -, with closed graph 272 VI-l.l -, with closed graph is Hausdorff 273

VI-1.4 Power set, of a set 10 0-2.7(1) - is an algebraic lattice 87 1-4.11 Power set functor, on CL 220IV-3.21 - preserves injective (surjective) maps

221IV-3.22 - preserves projective limits 221

IV-3.22 Priestly Duality, for distributive arithme-

tic lattices 325 VII-3.9 Prime element 701-3.11 -, in a lattice 701-3.12 Prime filter 731-3.17 -, in a power set 731-3.18 Prime ideal 73 1-3.17 -, in a distributive lattice 731-3.19,

751-3.22 -, in a poset 187IV-1.22 Prime ideals in C(X), closed, for X com­

pact Hausdorff 240V-1.8 Principal filter 2 0-1. 3 Principal filter embedding, on a poset

130-2.8(4) Principal ideal 2 0-1.3 Principal ideal embedding, on a poset

13 0-2.8(4) Pro-complete category 224 IV -4.1

INDEX

Pro-continuous functor, between pro­complete categories 224IV-4.1

Product, in a category 207 Remarks following IV-3.1

Projection maps, on a product 207 Re-marks following IV-3.1

Projection operator 21 0-3.8 -, on a continuous lattice 631-2.14 - preserves (directed) sups 220-3.11 - preserves (filtered) infs 22 0-3.11 Projective limit, in a category 208

IV-3.2 Projective limit cone, in a category 208

IV-3.2 Projective limit preserving functor, on

INFt 215IV-3.1Hf. Projective limit preserving functor 214

IV-3.10 Projective system, in a category 208

IV-3.2 Pseudoprime 751-3.23, 246 Remarks

following V-3.1

Quasicompact space 40 Remark pre­ceding 1-1.4

-, function space is a continuous lat­tice 13811-4.21

Random unit interval 64 Remarks fol­lowing 1-2.17

Regular open set, in a topological space 11 0-2.7(3)

Relation, auxiliary see: Auxiliary order -, closed 272 VI-I.1 -, converse 4 0-1.7 -, opposite 40-1.7 -, with closed graph 272 VI-I.1 Ring 120-2.7(7)

Saturated subset of a space 258 V-5.2 Scott-closed set 9911-1.3 -, characterization of 10011-1.4 Scott continuous function, defini-

tion 113 11-2.2 -, characterization of 11211-2.1 -, between algebraic lattices 112

11-2.1 -, between continuous lattices 112

11-2.1 -, form a complete lattice -, form a continuous lattice

11511-2.4 11611-2.8

INDEX

-, form an algebraic lattice 11611-2.8 -, form cartesian closed categories 117

11-2.10 - is always monotone 11211-2.1 - is jointly continuous on products

11811-2.13, 11611-2.9 - is separately continuous on pro­

ducts 11811-2.13 -, on complete semilattices 120

11-2.21 -, on up-complete posets 11911-2.18,

11911-2.19 Scott continuous function space, weight

of 1721II-4.12 Scott-open set 9911-1.3 -, chacterization of 10011-1.4 -, in a chain 10011-1.5(2) -, in a continuous lattice 10011-1.5(5) -, in a finite lattice 10011-1.5(1) -, in the square 101 11-1.5(6) Scott topology 9911-1.3 -, base for on continuous lattices 104

11-1.10 -, co-primes in 10511-1.11 - forms a continuous lattice 133

11-4.11, 134 11-4.14 -, functor preserves injective (surjective)

maps 219 IV-3.20 -, functor preserves projective limits

219 IV-3.20 - has a base of open filters 155

1II-2.11 - has enough co-primes 10711-1.14 -, in terms of nets 10211-1.6 - is a continuous Heyting algebra 134

11-4.16 - is a continuous lattice 10711-1.14,

1551II-2.13, 153 1II-2.2, 153 1II-2.3, 153 III-2.4

- is a dual Heyting algebra 134 11-4.15, 134 11-4.16

- is a function space 11611-2.7 - is a topological lattice 13411-4.16 - is an algebraic lattice 10811-1.15,

156 III-2.16 - is completely distributive 107

11-1.14 - is join-continuous 13411-4.15, 134

11-4.16 - is locally quasicompact 10611-1.13 - is order consistent 10911-1.17 - is productive 13311-4.11 - is sober 10611-1.12, 13411-4.14

369

- is super sober 310 VII-1.12 - is the upper topology 166 III-3.23 -, on a complete semilattice 108

11-1.16 -, on a continuous poset 10911-1.19 -, on a meet-continuous lattice 134

11-4.15 -, on an up-complete poset 108

11-1.16(2) -, on complete-continuous semilattices

10911-1.18 -, on up complete posets 139 11-4.23,

13911-4.24 -, on up-complete semilattices 139

11-4.22 -, primes in 10511-1.11 -, when hypercontinuous 333

VII-4.12 Self-acting monoid, in CL 235

IV-4.20ff. Semicontinuous function 140-2.10 Semilattice 4 0-1.8 -, algebraic see·: Algebraic semilattice -, complete see: Complete semilattice -, complete-continuous see: Com-

plete-continuous semilattice - has continuous and join-continuous

Scott topology 329 VII-4.4 - has continuous Scott topology 329

VII-4.4 - has small (open) semilattices 155

III-2.10 - has small semilattices 281 VI-3.1 -, ideals form an arithmetic semilattice

951-4.30 - is a compact semilattice 329 VII-4.4 - is a meet-continuous lattice and a

compact pospace 329 VII-4.4 - is a pospace if topological 275

VI-1.14 - is complete if compact 274 VI-1.13 - is meet-continuous if compact 274

VI-1.13 - is semicontinuous if topological 274

VI-1.13 -, meet-continuous see: Meet-con­

tinuous semilattice -, order connected 301 VI-5.13 -, semitopological 274 VI-1.11 see al-

so: Semitopological semilattice 274 VI-1.11

-, subinvariant pseudometric for 277 VI-2.1

370

-, topological 274 VI-1.11 -, topologically generating subset 243

V-2.2 -, topology is compatible if compact

27S VI-1.13 -, ultrametric for 288 VI-3.16 -, up-complete see: Up-complete se-

milattice -, when a compact pospace 328

VII-4.2 -, with continuous Scott topology 328

VII-4.2 Semilattice with small semilattices, cha­

racterization of 281 VI-3.3, 284 VI-3.7

-, closure properties of 281 VI-3.2 -, continuous morphisms between 282

VI-3.4 - is a continuous lattice if com­

pact 282 VI-3.4 - is embeddable in a cube 284 VI-3.7 -, quotient has small semilattices 283

VI-3.S Semitopological semilattice 270-3.23,

320-4.4,10911-1.17 - has the Scott topology 311 VII-1.14 - is a super-sober topological lattice

311 VII-1.14 -, local minimum in 300 VI-S.lO -, when topology is Scott topology 309

VII-1.9 Set, directed 2 0-1.1 -, filtered 2 0-1.1 -, free continuous lattice over 285 Re-

marks following VI-3.9 -, partially ordered see: Poset -, totally ordered see: Chain Sierpinski space 10011-1.5(3) -, is injective '122 11-3.3 Sober space 79 Remarks following

1-3.36 106 Remarks preceding 11-1.12

- and continuous posets 12711-3.18 - and up-complete posets 12611-3.16 -, dual to Heyting algebras 266

V-S.24 -, function space on 264 V-S.2 - is a continuous poset 265V-S.23 - is a monotone convergence space

12711-3.17 - is locally quasicompact iff O(X) is

continouous 259V-S.6

INDEX

-, topology is completely distributive 265 V-S.23

-, way-below relation in O(X) 259 V-S.6

Sobrification of a To-space 255 Remark following V-4.7, 257 V-4.9

Space, patch topology on 261 V-S.11 Spec: SUP i ~ TOp<'P, is left adjoint to

0: TOp"P ~ SUpi 254 V-4.7 Specialization order 123 11-3.6 - forms a continuous lattice 12911-4.S Stone-tech compactification, of a set

285 Remarks following VI-3.9, 290 VI-3.2S

Strict chain, in a complete lattice 194 IV-2.1

- satisfies the interpolation property 195IV-2.S(iii)

-, separate points in complete lattices 199IV-2.12

-, separate points in continuous lat­tices 200IV-2.16

Strong interpolation property see: Aux­iliary order

-, derived from auxiliary relation 202 IV-2.21

Sup map, has a lower adjoint 571-2.1 - is a homomorphism 30 0-4.2 - is Scott continuous 13311-4.13 -, on an up-complete semi lattice 64

1-2.18 -, on the ideals of a complete lattice

240-3.1S - preserves all infs 571-2.1 Sup preserving function 4 0-1. 9 Sup-semilattice 4 0-1.8 -, Dedekind cuts form a continuous lat­

tice 175 III-4.20 -, ideals form an algebraic lattice 88

1-4.12 Super-sober space 310 VII -1.10 -, is sober 310 VII-1.11 Supremum 1 0-1.1

To-space, all irreducible subspaces are Baire 267 V-S.27

-, all subspaces are Baire 267 V-S.27 - is sober 267V-S.27 - order-generates a continuous lattice

260 V-S.lO -, sobrification is locally quasicom­

pact 260 V-S.lO

INDEX

-, strict embedding in a locally quasi­compact space 260 V-5.1O

-, when O(X) is a continuous lattice 260V-5.1O

-, with O(X) continuous but not locally quasicompact 265 V-S.22

Topological space, cocompact topology for 312 VII-1.16

- forms a continuous lattice 130 11-4.10, 12811-4.4, 12911-4.6, 129 11-4.7

-, function space is a continuous lattice 12911-4.7

-, patch topology for 312 VII-1.16 -, saturated subset of 258 V-5.2 -, saturation of a subset 263 V-5.17 -, strict embedding of 260 V-5.7 Topologically generating subset, of a to­

pological semilattice 243 V-2.2 Totally disconnected space 94 1-4.25 Totally ordered set see: Chain

Ultrafilter, cluster points in the lower topology 164 III-3.16

-, in the lim-inf topology 159 1II-3.4 -, in the power set of a set 731-3.18 -, in the power set of a topological

space 74 1-3.20 Uitrametric, on a semilattice 288

VI-3.16 Unit, in a semilattice 40-1.8 Unit interval, is a continuous lattice 283

VI-3.6 Unital semilattice 4 0-1.8 Up-complete poset 150-2.11,330-4.8 -, algebraic 190 IV -1.34 -, closure operator on 190IV-1.32 -, from sober spaces 12611-3.16 -, Galois connection on 190 IV-1.34 - is a compact pospace 314 VII-1.22 - is super sober and quasicom-

pact 314 VII-1.22 -, kernel operator on 190IV-1.31 -, Lawson topology on 150 1II-1.21 -, lower adjoint preserves the way-be-

low relation 189IV-1.29 -, open filters in 78 1-3.32 -, open upper set in 77 1-3.30 -, order consistent topology on 108

11-1.16 -, Scott-closed subset 190IV-1.30 -, Scott continuous function on 119

11-2.18, 11911-2.19, 12011-2.20

371

-, Scott topology on 10811-1.16(2), 110 11-1.22,13911-4.23,13911-4.24

-, upper adjoint preserves directed sups 189IV-1.29

Up-complete semilattice 150-2.11,34 0-4.13,641-2.18

-, Scott topology is a Heyting algebra 13911-4.22

Upper adjoint 180-3.1 - is injective 200-3.7 - is surjective 200-3.7 - preserves directed sups 183

IV-1.12, 180 IV-l.4 - preserves prime elements 188

IV-1.26 Upper bound 1 0-1.1 Upper ~emicontinuous function 14

0-2.10 Upper set 2 0-1.3 Upper topology 10911-1.16 - is order consistent 109 II -1.17 - is the Scott topology 166 1II-3.23 Urysohn-Carruth Metrization Theo-

rem 276 VI-1.17 Urysohn-Nachbin Lemma 275 VI-US

Way-below relation 381-1.1,38 1-1ff. -, fundamental properties of 391-1.2 -, in a Boolean algebra 391-1.3(3) -, in a chain 391-1.3(1) -, in a complete distributive lattice 75

1-3.22 -, in a continuous poset 521-1.27 -, in a direct product 391-1.3(2) -, in a distributive lattice 75 1-3.22 -, in a finite lattice 401-1.3(4) -, in a meet-continuous lattice 41

1-1.5 -, in the lattice of ideals of a ring 40

1-1.3(5) -, in the lattice of open sets of a space

401-1.4, 741-3.20, 741-3.21 - is multiplicative 761-3.27 - satisfies the interpolation property

471-1.18 Way-below set, of an element 39 Re­

marks following 1-1.2 - is an ideal 39 Remarks following

1-1.2 Weak prime 81 Remarks following

1-3.40 Weak primes, order-generate 81 1-3.41

Zero, of a semilattice 4 0-1.8

Springer-Verlag Berlin Heidelberg New York

G. Gratzer

Universal Algebm 2nd edition. 1979.23 figures, 5 tables. XIX, 581 pages ISBN 3-540-90355-0

Contents: Basic Concepts. - Subalgebras and Homomorphisms. - Partial Algebras. -Construction of Algebras. - Free Algebras. -Indepedence. - Elements of Model Theory. -Elementary Properties of Algebraic Constuc­tions. - Free r. -Structures. - Appendices. -Bibliography. - Additional Bibliography. -Index.

This authoritative survey presents the state of the art of universal algebra, a field that has gained considerable importance since the publication of the first edition of this book in 1968. The main body of the book has been left unchanged; but seven new appendices a~d an extensive additional bibliography reflect the development of the last 10 years. Not all the appendices are by the author of the first edition; the other contributors are B. Jonsson, R Quackenbush, W Taylor, and G. Wenzel. " ... In the opinion of the reviewer, this book is suitable as a text for advanced graduate courses and seminars, and should be available in the library for use by faculty and students. It will certainly be, in the years to come, the basic reference on the subject"

American Mathematical Monthly

Springer-Verlag Berlin Heidelberg New York

H. H. Scheafer

Banach Lattices and Positive Operators

1974. XI, 376 pages (Grundlehren der mathematischen Wissen­schaften, Band 215) ISBN 3-540-06936-4

Contents: Positive Matrices. - Banach Lattices. - Ideal and Operator Theory. -Lattices of Operators. -Applications.

Infonnation: This is a comprehensive account of Banach lattices and positive linear operators (including positive finite matrices) with main emphasis on their relevance for and applications to measure theory, mean ergotic theory, operators between LP -spaces and spectral theory. Examples are given in each of the fifty sections, and the main text of each chapter is supple­mented by bibliographical notes and by exerCIses.