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Pacific Journal ofMathematics

INDUCED TOPOLOGIES FOR QUASIGROUPS AND LOOPS

KENNETH PAUL BACLAWSKI AND KENNETH KAPP

Vol. 45, No. 2 October 1973

PACIFIC JOURNAL OF MATHEMATICSVol 45, No 2, 1973

INDUCED TOPOLOGIES FOR QUASIGROUPS AND LOOPS

KENNETH BACLAWSKI AND KENNETH M. KAPP

The concept of a [semi] topological quasigroup is definedand the notions of induced groupoids and isotopes are extendedto the topological case. Necessary and sufficient conditionsare found in order for a continuously induced isotope of asemitopological quasigroup to be a semi topological quasigroup.

Given an injection ΐ of a topological space (A, Jάf) intoa set S acted on by a group, G, a topology ^~A on £ is intro-duced in a natural fashion under which i is continuous. WhenS=Q is itself a semitopological quasigroup and G is generatedby the left or right translations of Q the continuity or opennessof i can be checked by comparing the topology ^Ά withthat of Q. In particular this method is applied in § 3 to thestudy of topologically invariant subloops.

1* Preliminaries; continuously induced isotopes* We recall

(cf. [2]) that a quasigroup (Q, •) is a closed binary system i.e., agroupoid, where for any q e Q the left and right translations Xq, pq,both define bisections on Q. Thus given a quasigroup (Q, •) we candefine (cf. [2], [7]) the so called conjugate operations /, \ on the setQ by c/b = a iff ab — c iff a\c = b for a, b, ce Q. In the conjugatequasigroups (Q, /) and (Q, \) we will denote the left and right transla-tions determined by q e Q byλ£, p'q, X), and p) respectively, (In general,pi will denote the right translation determined by a relative to thebinary operation*,) A loop is a quasigroup with a two sided identity.A conjugate operation need not be associative; viz. if G — (Z, +),where Z is the set of integers then cjb = c — 6.

We now make the following definition (cf. [4], [5]):

DEFINITION 1.1. 1. A semitopological quasigroup [loop] is a triple(Q, , ά?~) where (Q, •) is a quasigroup [loop] and (Q, ~) a Hausdorffspace in which the left and right (multiplications) translations, λβ,pb, a, b e Q are homeomorphisms of Q onto itself. We will write Q =(Q, •; J ~) is a s.t.q. [s.t.L] in such cases.

2. A triple (Q, ^~) is a topological quasigroup [loop] if Q isa quasigroup [loop] and multiplication in Q and each of its conjugatesis continuous in both variables with respect to the same topology ^\We will write Q is a t.q. [t.i.] in such cases.

One makes the obvious genralizations of a [semi] topological group(s.t.g.) ([5], p. 28) to [semi] topological semigroup (s.t.s.) and groupoid(s.t.gd.). We observe that a s.t.q. is homogeneous and that a s.t.g.

393

394 K. BACLAWSKI AND K. M. KAPP

is also a s t q.It is easily verified that a topological quasigroup, Q, is a semi-

topological quasigroup; indeed each conjugate of Q is a topologicalquasigroup and hence also a semitopological quasigroup. Moreover atopological group is a topological quasigroup and conversely any topolo-gical quasigroup which is algebraically a group is a topological group.(For proof, one need only consider the appropriate translations in aconjugate quasigroup, e.g., λαλ« = 1Q while ghr1 = g/h in a group G.)

We now extend the definitions of topism, etc., in [6] to involvethe topological structure of given s.t.gd.'s:

DEFINITION 1.2. 1. Let (A, •; Stf) and (B, *; &) be two s.t.gd.'s.A triple of maps (τ; a, β) from A into B will be called a continuoustopism [isotopy] if 7, #, and β are continuous maps [and bijections]from A into B and (a? y)r — xa*yβ for all x9yeA. We will say that(7; tf, /9) is a topological isotopy if in addition 7, «, and /3 are home-omorphisms.

2. Let (A, •; jzf) be a s.t.gd. and μ, v continuous maps of A intoitself. The continuously induced groupoid {A, © ssf) = (A, μ, v; Jzf)of A is the groupoid in which χo y — χ^yv for each x,yeA. It μ and vare in addition bijections [homeomorphisms] we will say that (A, ois a continuously [topologically] induced (principal) isotope of (A,

It is easy to see that a continuously induced groupoid of a [semi]topological groupoid is itself a [semi] topological groupoid. Since it iswell known that any isotope of a quasigroup is a quasigroup one mightreadily expect that a similar result is true for continuous isotopes ofsemitopological quasigroups. The following example shows that thisneed not be the case even when one starts out with a topologicalgroup.

EXAMPLE 1.3. Let Q = (Q, +; J7~) where Q is the set of rationalnumbers and J7~ the topology inherited from the usual (open interval)topology of the real line. Let β — 1, the identity map, and define aas follows:

(a) a is the identity on Q Π ( V Ί , °o)(b) a maps Q Π (— °°, — τ/ΊΓ) continuously and strictly order

preserving onto Q Π (—•<*>, 0)(c) (0)α = 0 _ _ _ _(d) on [Qf±(-V2,j/2)]\{0}= [\J~=1 Q Π (V 2/(n + l), V 2/n)] U

[U^i Q Π (- V7 2 / ^ - i / 2 /(n +_1))] by mapping(i) Qn(l/2/(w + l),l/2/ίi) continuously and strictly order

preserving onto Q Π (τ/T/2w, τ/T/(2w - 1))(ii) ζ) Π (~τ/"2"/w, — l/~2/(n + ΐ)) continuously and strictly order

INDUCED TOPOLOGIES FOR QUASIGROUPS AND LOOPS 395

preserving onto Q n VΎ/(2n + 1), VΎβri).(We observe that such continuous strictly order preserving maps

do exist.)Thus a maps alternatively the disjoint covering of [Q Π

( - V % T/Y)]\{0^by {Ln = QΠ ( - ] / % -VΎ/(n + 1)), Rm = Q Π(1/ 2 /(m + 1), T/ 2 /ra)}«,«=i onto Q Π (0, V 2 ). One readily checks thatoί is a continuous bijection.

Now the continuously induced groupoid (Q, o j?~) = (Q, + α:, /3;^ ~ ) , which is an algebraic quasigroup, is not a semitopological quasi-group since the right translation <o0

o is not an open map in (Q, o J7~).We have for a G Q α° 0 = aa + 0^ = a" so that [Q Π ( - l / ϊ ,0 n [0, τ/"2") which is not open in (Q, ^ " ) .

THEOREM 1.4. Lβί (Q, o ? j^") = (Q, . α, /3; j^7") 6e α continuouslyinduced isotope of a s.t.q. (Q, J7~). If there exist b, ceQ such that

pb, pc are homeomorphisms then (Q, ° ^ ) is a topologically induced

isotope of {Q, ^Γ) and (Q, © ^~) is a s.t.q.

Proof. We will show that or1 is continuous and hence a homeo-morphism. Let be Q for which p? is a homeomorphism. Then fora? G Q: (x)ρϊ = χob = xabβ = (x)ap'bβ, h e n c e aρ\β = ft0. T h u s o r 1 = pb(pb)"1

and the first claim is verified. Dually /9"1 = λ^Ow0)"1 where λc° is thegiven homeomorphism.

In a similar manner one calculates that p° = <%plβ and λ° = /9λ*«for any ae Q. Since an isotope of a quasigroup is a quasigroup itnow follows that (Q, <> ^ " ) is s.t.q.

COROLLARY 1.5. Lei (Q, J7~) δβ α s.t.q. α^d (Q, o j^~) = (Q,• ί a, β; j/~) be a continuously induced isotope of Q. Then the fol-lowing are equivalent:

1. (Q, o ^Γ) is a s.t.q.2. there exists b, c eQ such that λc° and to6° are homeomorphisms.3. (Q, o ^ " ) s α topologically induced isotope of Q.

Example (1.3) shows that if (Q, °) is not a loop then (Q, o ^ " )need not be s.t.q. However the following corollary is valid since theloop identity e determines the homeomorphisms λj = p°.

COROLLARY 1.6. // a continuously induced isotope of a s.t.q. isa loop, then it is a s.t.l.

We comment in conclusion that many of the results of [6], §2have topological analogues.


2* Translation topology; embeddings of semitopological quasi-groups* The algebraic problem of when a binary system of one typecan be embedded in a binary system of another type has in manycases been solved. Difficulties arise, however, when the algebraicsystems have in addition a topological structure. Then there is theadded problem of finding an embedding which is also a topologicalhomeomorphism. Indeed, it is also often desirable to have the imageopen in the second space. Solutions are then dependent on the rela-tionship between the topologies of the two structures. This problem,it will be seen, includes the more limited one of comparing with thegiven topology of the image space a topology induced by the giveninjection from the topology of the domain space.

DEFINITION 2.1. 1. An ordered 5-tuple (i; A, j*f\ S, G) is calleda system for an induced topology of S if

1. (A, s*f) is a topological space,2. G is a group acting on the nonempty set S and3. i is an injection of A into S.In such a case we will say that (i; A, s*f\ S, G) is an s.i.t. for

brevity.

2. Let ^~A = {BsSKBgtf-1 e j y for each g e G}. Here (Bg)i~ι ={a e AI (a)i e Bg). JίΓA is called the induced action topology with re-spect to the s.i.t. (i; A, j ^ ; S, G).

One readily checks that ^~A is a topology on S; it is not neces-sarily Hausdorff even when A is Hausdorff. If A £ S and i is theinclusion map, we see that ^"A — {B g S\Bg Π Ae s^f for each g e G}.

EXAMPLES 2.2. 1. Let {A, sf) be a discrete space. Thenfor any s.i.t. is also discrete.

2. Let (i; A, stf) S, {e}) be a s.i.t. Then j r ; = {B S S \ Bi~ι eThus ^ A contains the set of all subsets of S\(A)i as well as {(U)ί\ Ue

Whence (S, S~A) is Hausdorff precisely when (A, j y ) is Hausdorff.

3. Let (i; R, &; C, G) be a s.i.t. where (R, &) is the real linewith the usual topology, C the set of complex numbers, i the naturalinclusion map of R into C and G the group of all bijections on COne checks that άΓR = {jBg C\C\B is finite or B = 0}. Clearly ^~R

is not a Hausdorff topology for C.One sees immediately that when (i; A, Jϊf; S, G) is a s.i.t. then

i: (A, Ssf) ~> (S, J7~A} is continuous and each g e G determines a homeo-morphism of (S, ~A) onto itself. Thus if G acts transitively on Sthe space (S, ~A) is homogeneous. Moreover Jf~A is the strongest


topology on S under which the map i o g: A —> S is continuous for eachgeG.

Following Bruck ([2], p. 54) we make the following definition:

DEFINITION 2.3. Let (Q, •) be a quasigroup. We set ^y£9 — /ί9{Q),^/^ — £'λ(Q) and ^€ = ^/?(Q), the groups, respectively, generatedby the right, left, and both right and left translations of Q. Thus,e.g., ^fp is generated by {pg\geQ}

It is clear that for a quasigroup (Q, •) the three groups of transla-tions are transitive groups acting on Q. Thus if (i; A, j y ; Q, G) isa s.i.t. where G — /£{Q) \^9{Q) or ^fλ{Q)\ we will speak of the[right, left] translation topology, J7~A, of Q.

THEOREM 2.4. Let (Q, Jf) be a s.t.q and (i; A, s*f\ Q, G) a

s.i.t. where G = ^£{Q) [^//9{Q), ^fχ{Q)]* Then1. i is continuous iff j ^ ~ £ J7~A

2. if i is open then J7~ 2 ^7~A and3. if i is an open embedding then ^~ -

Proof. If J7~ £ άΓA then i is clearly continuous. Conversely,suppose i is continuous. Let B e ^T Since for each geQ, pg, Xg arehomeomorphisms, G is in each case a group of homeomorphisms onQ. Thus if teG, (B)t = Bte^Z Whence by the continuity of i,{Bt)i~ι G j y for each t e G and thus 5 e j r ; , ^~ <^^~A and the proofof 1 is complete.

Let us now suppose that i is a ^ o p e n map. Let S e , / ] , 2? 0 .Let be B. We will find a ^^open neighborhood of δ contained in B.Suppose G = ^J(Q); the proofs for the other cases are analogous.Let a G (A)i. There is a unique g G Q such that 6g = a. Now (α)i"1 e(Bq)i~ι G j ^ by definition of ^~A. Since i is open a e {{Bq)i~ι)i = U =Bq Π (A)i G ^ I Since /off is a homeomorphism in (Q, ^ " ) it followsthat b = (a)(Pqr

le(U)(pqri = (Bqn(A)i)(pq)~1SB, and (C/)^)-1 is

the desired ^ o p e n neighborhood. Whence B e ^~ and <^~A £ ^TThe third result is an immediate consequence of the first two and

the proof is now complete.From the results following (2.2) we now see that when (i; A,

Q, G) is a s.i.t. where Q is a quasigroup and G = ^£{Q), if (Q,is a Hausdorff space then (Q, j?^) is a s.t.q.

3* Semitopological loops* We will now apply the concepts ofaction and translation topologies to the study of invariant subloopsof a loop.

The first two parts of the following definitions can be found inBruck ([2], p. 61).


DEFINITION 3.1. 1. Let (Q, )be a loop with identity e. Thesubgroup of ^f defined by J^ = *J*~(Q) = {te^€\(e)t = e} is calledthe inner mapping group of (Q, •) and we note that ^ is thestabilizer of e in

2. Let (Q, •) be a loop and (if, •) a subloop of ζ). We say thatH is a normal (invariant) subloop of Q if (H)<J^ S £Γ and we writeIΓ<3 <? in such cases.

3. Let (if, Sίf) be a topological space and a subloop of (Q, •)•We will call fl" = (if, Sίf) a topologically invariant subloop of Q iffor each Ke Sίf and i e ^ iζ? 6 ^ We write #<!* Q in such casesand note that if H^tQ then

THEOREM 3.2. Let (H, \3ff) be a s.t.L and suppose H<\Qwhere (Q, •) is loop. The followings are then equivalent:

1. -BΓ<*Q2. Jg^ £ ^7/ where J7~H is the action topology generated from the

s.i t. (i; fl, ^T; Q, ^T(Q)).3. There is a topology J^Γ for Q under which (Q, j^7~) is a s.t.L

with (H, £%f) as a subspace.4. There is a topology ^ for Q under which (Q, J7~) is a

s.t.L with (H, J%f) as an open subspace.

Proof. Assume 1. H^tQ. (^T = ^(Q)) is defined in (2.3)).Let Ke^ and te^f. We will show that Ktf] He^έf. We assumefor the sake of argument that Kt Π HΦ 0 . If he KtΓ\ H then h =(k)t for some ke K. It is easily checked that pk°t° {ph)~ι e J?. Whence(H)ρk o t © do^)"1 g H and (Jϊft)ί g iίp^. It follows, since H is a closedsubloop that (#)£ S H. In particular ίΓί s H and Kt f] H = Kt.

Now let m = (e)ί. Then top~ιej^ so that {K)to p~ι = {Kt)ρ^e£%f. But HtξΞ:H, a s.t.ί., thus meH and ^ w is a homeomorphismon if. It now follows that Kteβέfί Whence if 6 J ^ , and 1 -> 2. (Wealso note that if G is any subgroup of ^ then £ίf gΞ ^^ where^ ^ is generated from the s.i.t. (i; H, £ίf; Q, G).)

Assume 2 now holds, i.e., 3ίf £ ^ ^ . Then i: (if, <%*) ~> (Q, ^ , ) ,which is continuous by Theorem (2.4), is an open embedding. Thuswe need only show that ^ ^ is a Hausdorff topology for Q and 4 willfollow. Thus let a, be Q with a Φ b. If Ha Π Hb — 0 we are donesince Ha, Hbe^H. But if Ha Γ\ Hb Φ 0 then iϊα = Hb since JBΓ<3

Q. Thus a = hb for some heH\{e}. Now (iϊ, ,^^) is Hausdorff sothere are disjoint sets A and B such that ee Be Sίf and he AeIt follows that beBb and α 6 Ab. But Aδ n 5& = 0 and Aδ, ΰδsince ^ is a homeomorphism. Whence (Q, ^ ) is Hausdorff and the


proof of 2 —* 4 is complete.Clearly 4 —> 3. We will now prove the implication 3 —> 1.Assume 3 holds and let je J^{Q) and Ke 2ίf. Since H<\Q we

have Hj = H. Now K=HnU for some Ue j Π Thus Kj = (H n U)j =Hj Π ί7i = H Π C#. Since j" is a homeomorphism of Q, Uj e ^T Henceiζ? e ^ ^ and the proof of this implication and the theorem is complete.

COROLLARY 3.3. Let (Q, J7~) be a s.t.i. The component E, ofthe identity, β, with the restricted product and inherited topology isa topologically invariant subloop of Q.

Proof. We need only show that E<\Q. Connectivity is pre-served under continuous maps so that for each te^£{Q), Et is con-nected. Thus either Et Π E = 0 or Et £ E. In the latter case E gΞ# r ι and since r 1 e ^f{Q) it also follows that Et~ι £ #. Whence # =J?ί = M"1. Now for any g, he E we have ge EΠ -E^ and he E f]EXh. Thus £7 = 1?^ = E o-1 = ί/λ = EXJ1 and it follows that E isclosed. In particular for any j e *J^(Q), e = (e)j, E Π E Π Eά Φ 0 sothat 2?i = E and hence E ^ Q. The result now follows from Theo-rem (3.2).

DEFINITION 3.4. Let (iϊ, £έf) be a s.t.ϊ. and a subloop of aloop (Q, •)• Then RH = {Bt\Be £έf,te ^P{Q)} and LH = {Cs\Ce£έf,s e ^C(Q)} are called, respectively, the right and lest coset coveringsof Q with respect to (H,

THEOREM 3.5. Let H<\t Q where (H, ^f) is a s.t.l. then Rπ =LH is a subbase for a topology &H = J*fH under which (Q, •) is as.t.i. Moreover if i is the natural inclusion map of H into Q andG = t(Q)[^/tP(Q), ^C(Q)] then ^H = ,^H = fH, where H is generatedfrom the s.i.t. (i; H, £έf\ Q, G), and i embeds (H, C9ίf) as an opensubspace of (Q,

Proof. Let ΰ e ^ T and te^fp{Q). Set m = (e)t. Then toX^s^ so that (B)toχ-1 e £ί? since H^t Q. Thus Bt = {{B)t o λ-:)λm e Z^and i?^ S L^. Similarly LH £ i?^ and equality now follows. ClearlyU RH = Q so that iϋ^ is a subbase for a topology on Q.

We will show that (Q, ^ ) is Hausdorff. Let qtφ q2e Q. SinceH<\Q (cf. [2], pp. 60, 61) either Hqλ = ίZg2 or ^ Π Hq2 = 0 . In thelatter case, since Hq19 Hq2 e RH and e e H we have two disjoint openneighborhoods in &H of qt and q2. Suppose now that Hq1 = ί?g2.Then gr = hq2 for some fee H\{e}, where e is the loop identity. Since(H, £ίf) is Hausdorff there are open disjoint neighborhoods, A, Be


of e and h respectively. Then Aq2 Γ) Bq2 = ( 4. Π B)q2 = 0 , g2 e 4g2

and ςfi e Bq2. But A#2, J5g2 G RH and it follows that (Q, ?H) is Haus-dorff.

Now let te^fpiQ). Then if f|*-BA i s a basic open set of ^g^we have (Γ) i?A)£ = Π (.£?*£<) ί = Π B^Ut) another basic open set of &H%

Thus t is an open map. Since t~ι e ^£9 it is also true that t is con-tinuous. Whence t is a homeomorphism. Since all right translationsdetermined by q e Q are in ^£p it follows that each such pq is ahomeomorphism. In a similar fashion since RH = LH, each λg is ahomeomorphism. Thus (Q, ^ ^ ) = (Q, .2^) is a s.t.q.

We have already seen in the proof of Theorem (3.2) that έ%f gwhere ^~a is generated from the s.i.t. (i; H, £ίf\ Q, G) and G =

^ ] . Let Bt e RH where B 6 £έf, t e ^£p. Since t is a homeo-morphism and Sίf S ^ Ί r it follows that Bt e <ί7"H and thus &H £ ^~π.Conversely since the inclusion map i\H—+Q is open for the s t l(Q, &H) by Theorem (2.4.2) we have &H 2 ^ ^ and equality follows.The other assertions are now clear and the proof is complete.

The following corollary is now an immediate consequence of theabove and Theorem (2.4).

COROLLARY 3.6. Let (H, •; <%f) and (Q, ^y) be two s.t.Z.'s. withH<ίt Q. If i is the natural inclusion map of H into Q then

1. i is continuous iff y £ &H

2. i is open iff^"^ &H

3. i is an opening embedding iff

We conclude with an example which illustrates both the existenceof a normal subloop of a s.t.ί. which is not topologically invariant.The example will show that necessity of the topological invariancein the hypothesis of several results viz. Theorem (3.2), and Theorem(3.5).

EXAMPLE 3.7. Let C be the set of complex numbers. For me

(Z2, +) we define an action of Z2 on C by z~ — I- ?Λ m ~ « where ze\% I I ΊYV -— X

C and z denotes the conjugate x — yi — x + yi = z (x, y e R, the reals).It is easy to check that this is an action and that (zλ + z2)™ — zψ + zψ.We will consider R, the set of real numbers, with two differenttopologies: ^p, the usual topology of the real line, and £f theSorgenfrey topology (generated by half open intervals). We can nowdefine the following two topologies on C where we make the usualidentification x + yi—+(x, y). The usual topology generated from & x& will be denoted by ^ while we will denote by &' the cartesianproduct topology generated from & x SZ It is readily checked that


(C, + ; ^ ) and (C, +;&') are both semitopological groups.Let P = C x Z2. We construct a group on P by defining a binary

operation as follows:

(w, m) + (z, w) = (w + 2-, m + n) for (w, m), (z, ri)e P .

A straightforward check shows that (P, + ) is a group whichcontains the subgroup C x {0}. One readily checks, for example, that(z, m) + (-Z-, m) = (0, 0) for any (2, m)eP and that (0, 0) is theidentity of P. We identify now C and C x {0} since they are isomor-phic under the obvious natural map. It is easily seen that [P: C] =2 and thus C^P. Furthermore (z, 0) + (0, 1) = (z, 1) Φ (z, 1) = (0, 1) +(z, 0) so that P is not abelian. However (z, 0) = (0,1) + (z, 0) + 0, 1) =(z, 0)ΐ(0,i). Thus, the inner automorphism determined by (0, 1) whenrestricted to C is just conjugation on C. Now if we consider thesemitopological loop (C, + ;&") as a subloop of (P, +) we see that(C, + W) is a normal but not a topologically invariant subloop, sinceconjugation in (C, ' ) is not a continuous map.

Let (i; C, if"; P, ^ # ) be a s.i.t. We shall show that the topologyinherited by (G)i from J7~o is exactly <g*. It will then follow that iis not open or even an embedding so that the converses of (2.4.2)and (2.4.3) are false when we take (P, +; J7~c), i.e., J7~ = J7~c, in thehypothesis. Now let ϋΓg P. Since P= Cϋ[C + (0, 1)] we can de-compose K into two subsets K — Kx U [^2 + (0, 1)] where Kt S CSince (P, +) is a group ^€(P) = {XqiρQ2\qie P) and ίΓe ^ iff [(w, m) +iΓ+ («, n)\ Γ\Czrέ?'. But (w, m) + K + (s, w) = (w + Kψ + z-, m + ) U(w + Kτ + «i±s, 1 + m + w). Thus if we intersect with C = C x {0}we get either w + z- + JK"? when m — n i.e., if m + w^O or w + z^^ + Kψwhen mφ n i.e., if m + n = 1. Hence Ke ^"c iff Kl9 Ku K2, K2 e ^ '(note: (C, + ; ^ " ) is a s.t.g.!) It then follows that Kγ and iΓ2e ^Whence a subset if £ P is open iff i£ = Kx (j [i^2 + (0, 1)] where if e^ . Clearly (P, ^ ) is Hausdorff and it follows that (P, + ^~c) isa s.t.g.

The inclusion map ί: (C, +;&") —• (P, +lJ7~c) is continuous butnot open since the inherited topology of (C)i is ^ and ^ £ ^ ' .Indeed, i is not even an embedding (cf. (2.4)) as we were to show.Theorem (3.2) is thus false if topologically is omitted from the hy-pothesis since here cg" % J7~c.

We now remark that Theorem (3.5) does not hold in this example.If K is the upper half plane including the real axis then ϋ T e ^ ' .However, if K + (0, 1) = (w, m) + L for some Le^' then (—w-, m) +K + (0, 1) G <^'. Hence m + 1 = 0 i.e., m = 1 and we have, sinceK = K x {0}, (-w, 1) + JBL + (0, 1) = -id + K which is not in <g".Theorem (3.5) fails for in this example neither &c nor ^G are topologies


under which (P, +) is a s.t.g.: the inner automorphism i(0>1) is notcontinuous.

REFERENCES

1. J. Aczel, Quasigroups, Nets, and Nomograms, Advances in Mathematics, 1 (1965),Academic Press, 383-450.2. R. H. Bruck, A Survey of Binary Systems, Ergebnisse der Math. Neue Folge, Vol.20 Springer, 1958.3. K. H. Hofmann, Topologische loops, Math. Z., 70 (1958), 13-37.4. K. H. Hofmann and P. S. Mostert, Elements of Compact Semigroups, Merrill Books,Inc., Columbus, Ohio, 1966.5. T. Husain, Introduction to Topological Groups, W. B. Saunders Co., Philadelphia,Pa., 1966.6. K. Kapp and K. Baclawski, Topisms and induced non-associative systems, Pacific J.Math., 36 (1971), 45-54.7. S. K. Stein, On the foundations of quasigroups, Trans. Amer. Math. Soc, 85 (1957),228-256.

Received August 30, 1971 and in revised form September 14, 1972.

HARVARD UNIVERSITY

AND

UNIVERSITY OF WISCONSIN-MILWAUKEE

PACIFIC JOURNAL OF MATHEMATICS

EDITORS

H. SAMELSON J. DUGUNDJI

Stanford University Department of MathematicsStanford, California 94305 University of Southern California

Los Angeles, California 90007

C. R. HOBBY RICHARD ARENS

University of Washington University of CaliforniaSeattle, Washington 98105 Los Angeles, California 90024

ASSOCIATE EDITORS

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Pacific Journal of MathematicsVol. 45, No. 2 October, 1973

Kenneth Paul Baclawski and Kenneth Kapp, Induced topologies for quasigroupsand loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393

D. G. Bourgin, Fixed point and min−max theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 403J. L. Brenner, Zolotarev’s theorem on the Legendre symbol . . . . . . . . . . . . . . . . . . . . . 413Jospeh Atkins Childress, Jr., Restricting isotopies of spheres . . . . . . . . . . . . . . . . . . . 415John Edward Coury, Some results on lacunary Walsh series . . . . . . . . . . . . . . . . . . . . 419James B. Derr and N. P. Mukherjee, Generalized Sylow tower groups. II . . . . . . . . 427Paul Frazier Duvall, Jr., Peter Fletcher and Robert Allen McCoy, Isotopy Galois

spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435Mary Rodriguez Embry, Strictly cyclic operator algebras on a Banach space . . . . 443Abi (Abiadbollah) Fattahi, On generalizations of Sylow tower groups . . . . . . . . . . . 453Burton I. Fein and Murray M. Schacher, Maximal subfields of tensor products . . . 479Ervin Fried and J. Sichler, Homomorphisms of commutative rings with unit

element . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485Kenneth R. Goodearl, Essential products of nonsingular rings . . . . . . . . . . . . . . . . . . 493George Grätzer, Bjarni Jónsson and H. Lakser, The amalgamation property in

equational classes of modular lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507H. Groemer, On some mean values associated with a randomly selected simplex

in a convex set . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525Marcel Herzog, Central 2-Sylow intersections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535Joel Saul Hillel, On the number of type-k translation-invariant groups . . . . . . . . . . 539Ronald Brian Kirk, A note on the Mackey topology for (Cb(X)∗, Cb(X)) . . . . . . . . 543J. W. Lea, The peripherality of irreducible elements of lattice . . . . . . . . . . . . . . . . . . . 555John Stewart Locker, Self-adjointness for multi-point differential operators . . . . . . 561Robert Patrick Martineau, Splitting of group representations . . . . . . . . . . . . . . . . . . . 571Robert Massagli, On a new radical in a topological ring . . . . . . . . . . . . . . . . . . . . . . . 577James Murdoch McPherson, Wild arcs in three-space. I. Families of Fox-Artin

arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585James Murdoch McPherson, Wild arcs in three-space. III. An invariant of

oriented local type for exceptional arcs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599Fred Richman, The constructive theory of countable abelian p-groups . . . . . . . . . . 621Edward Barry Saff and J. L. Walsh, On the convergence of rational functions

which interpolate in the roots of unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 639Harold Eugene Schlais, Non-aposyndesis and non-hereditary

decomposability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 643Mark Lawrence Teply, A class of divisible modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . 653Edward Joseph Tully, Jr., H-commutative semigroups in which each

homomorphism is uniquely determined by its kernel . . . . . . . . . . . . . . . . . . . . . . 669Garth William Warner, Jr., Zeta functions on the real general linear group . . . . . . . 681Keith Yale, Cocyles with range {±1} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693Chi-Lin Yen, On the rest points of a nonlinear nonexpansive semigroup . . . . . . . . . 699

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1BDJGJD +PVSOBM PG .BUIFNBUJDT · INDUCED TOPOLOGIES FOR QUASIGROUPS AND LOOPS KENNETH BACLAWSKI...

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Transcript of 1BDJGJD +PVSOBM PG .BUIFNBUJDT · INDUCED TOPOLOGIES FOR QUASIGROUPS AND LOOPS KENNETH BACLAWSKI...