T H E E M B E D D I N G O F A D I S T R I B U T I V E L A T T I C E 1)

I N T O I T S I D E A L L A T T I C E I S P U R E

EVELYN NELSON

135

A slightly stronger result than that announced in the title, namely that the 'canon- ical' embedding of every distributive lattice into its ideal lattice (under which each element is mapped to the principal ideal it generates) is super-pure (for definitions of pure and super-pure see w is a corollary of the following:

THEOREM 1. For every lattice embedding (o of a finite distributive lattice D into the ideal lattice of a distributive lattice L, there exists a lattice embedding ~b: D ~ L such that

(*) For all x , y~D, ~k(x)eq~(y) if f x<~y, and, i f ~b(x)=(a], the principal ideal generated by a, then ~ (x) = a.

This theorem, without the restriction on ~O pertaining to principal ideals explicitly stated, was proved by H. Gaskill [5].

The result of the title is not true with 'distributive' deleted; w 3 contains an example of a (non-modular) lattice whose embedding into its ideal lattice is not pure.

! would like to thank Professor R. Wille for helpful conversation concerning the lattices in w 3.

w 1. Distributive lattices

The proof of Theorem 1 is more or less implicit in Gaskill's proof 1-5] ; however, since the original version, as it appeared in print 1-4], is incorrect, a proof is included here.

For this section, let D and L be distributive lattices, D finite, and ~b:D ~ I(L) a lattice embedding of D into the ideal lattice of L.

Let J and M be, respectively, the set of join-irreducible and meet-irreducible elements of D, and for xeD, let J r = {yeJ[y<~x}, Mr= {yeM]y>>,x}, and h(x) be the height of x in D. Note that x = V J~ = A Mr. (See Birkhoff [2], for definitions and basic results).

It is easy to check that an order-preserving map r ~ L is a join-homomorphism iff, for all xeD, r (x)= V r (Jr).

LEMMA 1. Suppose that ~Po:D ~ L is a (set) mapping with property (*) for all x, y ~ D, ~o (x) e q~ (y) iff x <~ y and, / f ~b (x) = (a] then ~ o (x) = a,

and that ~ : D ~ L is a meet-preserving map with the following properties:

1) Research supported by the National Research Council of Canada.

Presented by H. Lakser. Received May 14, 1973. Accepted in final form August 2, 1973.

136 EVELYN NELSON ALGEBRA UNIV.

(el) v (4)/or a l lxEm (P2) ~b o (x) ~< V ~b (J~) and tp ( x ) ~ (x)for all xED.

Then if a~D is maximal such that V ~b(Jo)<~O(a), there exists a meet-preserving ~*:D ~ L with properties (P1) and (P2) such that ~O* (a)= V ~0" (Ja) and, for all xeD with h(x)>~h(a), /f ~(x)= V ~k(Jx) then ~b*(x)= V ~b*(Jx).

Proof. Let b = A {x~D [x~.a}. Since ~b(b)~q~(b)= V ck(Jb) and L is distributive, there exists bz~q~ (z) for each z~Jb such that V {bz [z~Jb} =q; (b).

Then the map ~ : D ~ L given by ~(x)=~b(x)v V {b~ [ZeJb, z<<.X} is order- preserving, ~(x)<~(x)~4~(x) for all xeO, and ~ ( x ) = r ifx>.b.

Define ~*: D ~ L by 4,*(x)= A { V r

Then ~b* is meet-preserving. Note that if x~J then ~b*(x)>~ V ~(J~)=~(x). Consequently, for all xeD,

V ~*(Jx)~ > V r > V ~k(J~)>~Oo(X); since we also have~b* (x)~[") {(o(y) ly~M~} = q5 (x), it follows that ~b* has property (P2).

Since for all z~D, ~k*(z)~> V ~(Jz), it follows that ~b*(x)= V r < V ~b*(Jx) for all x~M, i.e. ~b* has property (P1).

If x>_.b then ~ ,* (x )=A { V ~b(J,)v~(b)lY~M~}= A {~k(y)v~b(b)ly~M~} = A {~/(y) ly~M~}=~k(x). In particular ~b*(a)=tk(a).

If x~J and x.~a then a < x v a and consequently, by the maximality of a, ~b (a) ~< q~ (x v a) = V ~b (J~ v,) = V ~b (J~) v V ~O (J,) = ~b (x) v V ~O (3",). Since for every y~a there exists xeJ~ with x.lza, it follows that ~ ( y ) v V ~k(Jo)>~(a) for all y~za, and thus ff*(a)=~b(a)~< A {~b(y)ly~za}v V ~b(J,)=~O(b)v V ~b(J~)= V ~(J~)~< ~< V ~b* (J,); since the reverse inequality holds trivially, we have ~b* (a)= V ~b* (Jo).

Finally, ifh(x)>_.h(a) then x>_.b; if in addition ~k (x)= V ~' (J~) then V ~,* (J:,)>-- /> V r V ~k(J~)v~(b)=~O(x)vq;(b)=~O(x)=r and thus ~b*(x)= = V ~b* (J~). This completes the proof.

Proof of Theorem 1. For each pair (x ,y )eDxD such that x~gy, choose a~ye~b (x)-~b (y) such that, if ~b (x)= (a] then a~=a for all y~x .

Then ~b0:D ~ L defined by

~'o(X) = V {a~, lye.x}

has property (,), and moreover, if r/:D + L is any map such that ~ko (x)~<r/(x)e~b (x) for all x~D then q also has property ( . ) and hence in particular is an embedding.

For each pair (x, y)~ O x O with x ~ J~, choose b~e ~b (x) such that V {b~,~ I x e J~}/> I> ~b o (y), and define ~O~ :D ~ L by:

~kt (x)={~bo (x) if xCJ ~Oo(x)vV{b~,lx<<.y } if x~S

Then, for all x~D, ~ (x)egp (x) and V ~O~ (Jx)>_.~bo (x).

Vol. 4, 1974 THE EMBEDDING OF A DISTRIBUTIVE LATTICE 137

Define ~2:D ~ L by ~b 2 (x)= V { ~ (Y) IY ~<x}; then ~O 2 is order-preserving, and for all x~O, ~1 (x)<~ ~z (X)~dP (x).

Now, ~: D ~ L given by

r A { V Cz(Jy)lY~M~,}

is meet-preserving, and O2(z)~<~b (z) if zEJ. Furthermore, for all x~D, V O(Jx)>~ i> V ~O2(Jx)~> V Ol(J~)>~bo(X), and i fxeM, then O(x)= V O2(J~,)~< V ~b (J~) and so (x)= v (:A.

But now, since D is finite, a simple induction argument, together with Lemma 1, yields a meet-preserving ~b* :D ~ L such that for all xeD, ~ho (x)<~ ~k* (x)eqb (x) and ~b* (x)= V ~h* (J~); since the latter implies that if* is join-preserving this completes the proof.

w Purity and super-purity

For an arbitrary (universal) algebra A in an equational class A, let A ['X] be a free extension of A (in A) by the set X of indeterminates, where X is a countable set. A homomorphism f : A ~ B in A is a pure embedding iff every finite subset of A IX] 2 which is contained in the kernel of some homomorphism A I-X] ~ B over f is also contained in the kernel of some homomorphism: A I-X] ~ A over A. (See B. Bana- schewski and Evelyn Nelson ['1]).

A homomorphism f : A ~ B is called a super-pure embedding iff, for every finite set Xo, every finite subset F~_A I-Xo-I 2 which is contained in the kernel of a homomor- phism A [-Xo] ~ B over f which is 1-I on Xo is also contained in the kernel of a homo- morphism A [Xo] ~ A over A which is 1-1 on X0.

(As pointed out to me by P. D. Bacsich, it is easy to see that this condition is equivalent to: for every pair F, G of finite subsets of A I-X] 2, X arbitrary, if there is a homomorphism h:A IX] ~ B over f with F _ K e r h and G _ C Kerh ( = A IX] z - - K e r h ) , then there is also such a homomorphism A [ X ] ~ A over A, i.e., in the terminology of Bacsich, Alg. Univ. 2 (1972), 354-360, a homomorphism is a super- pure embedding iff it is existential.)

Note that every elementary embedding is super-pure, and every super-pure embed- ding is pure, but neither of the converse statements are true; the embedding of the lattice N of natural numbers with the usual order into its ideal lattice, I(N), is super- pure by the result below, but is not elementary since I(N) satisfies (3x) (Vy) (y~< x), N clearly does not. Moreover, in w 3 we give an example of a modular lattice whose embedding into its ideal lattice is pure but not super-pure.

THEOREM 2. In an equational class of finite type in which the finitely generated

138 EVELYN NELSON ALGEBRA UNIV.

algebras are finite, an embedding f : A ~ B is super-pure iff , for every finite subalgebra C c_ B there exists a monomorphism g: C ~ A such that g of [ f - 1 (C) is the identity map.

Proof. Suppose f:A---, B is super-pure, and that C=__B is a finite subalgebra. Let Xo = {xc [ c~C} be a set with the same cardinality as that of C which is disjoint from A and B, and let Fc_A [Xo] 2 be

{ ( f (xc,, ..., xc.), x:(c~ ..... ~.)) [ c 1 . . . . , c ,~C, f an n-ary operation} w

u {(x . c) l C}.

The super-purity o f f yields a homomorphism h:A [Xo] ~ A over A which is 1-1 on Xo such that F _ Kerh; then g: C ~ A given by g (c) = h (x~) is the desired monomor- phism.

The converse is trivial.

COROLLARY. The embedding of a distributive lattice into its ideal lattice is super-pure.

Proof. This follows immediately from Theorems 1 and 2.

w (Counter-)examples

The lattice L of Figure 1, the dual of which appears in Wille [-6] as an example of an infinite lattice of finite width which is generated by three elements, has the property that its embedding into its ideal lattice I (L ) (see Figure 2) is not pure. This can be seen as follows: If u is the unique co-atom of L, then

F = { ( x v u , 1), ( y v u , 1), ( x v y , 1), (XA y, XAU), (XA y, yAU)}~_L[x , y ]

is contained in the kernel of a homomorphism L Ix, y] ~ I (L ) over L but not in the kernel of a homomorphism L Ix, y] --+ L over L. Note that L is not modular.

II I \~

11~\\ / 11 ~\

\

I(L)

Figure 1. Figure 2.

Vol. 4, 1974 THE EMBEDDING OF A DISTRIBUTIVE LATTICE 139

The lattice Lo of Figure 3 is a modular lattice whose embedding into its ideal lattice is pure but not super-pure. Lo is the dual of FM(J4~) in the terminology of Day, Herrmann and Wille [3]; its ideal lattice is diagrammed in Figure 4. L o is a retract of its ideal lattice (the map qS:I(Lo)--*Lo which maps L o identically, such that

/ / / z

a \ \

"-\ \ \ \ \ x \ \ k \

X \ \ \ x X \ x ~ '

\ \ \ \ \X %\N

.," / \ \ "x .

" / , ' % . " ~ ' ~ d ,, ,, , , . , ~ , -,,o / / / / z / / / / ~////

" / / / / / ' / / / . , "b, Cn / / / z / . / ./- ~ / " / / z/ / / / /

/ / / / / / / / /

/ Lo

Figure 3.

1

bn,x~ bf\, ~ ' ~ ' x "~'k 5 cs,'x,. ,''h,,c? /d , , , a X " x " ~ a f , " " . , , ' .~r.. ,"-. .c~ , ~o

\X X k \N / / / / / \ \ \ \ \ \ x\ \ \ / / / / / / / / / /

x N \ \ \ x -, 1/ / / / / / / \ x \ \ \ x / / / / / \ \ \N\ \ \xx\ k ,." / / / / N\

\X / / / / \ \ \ \ \ \ \ \ x )~ ./ / /

,\ "" / / / / / / / /

/ / ~ / / \ \ \ \ \ \ / / j / / /, l / X\ \N / / / /

\ / / \ /

l(Lo) Figure 4.

140 EVELYN NELSON

~b(d,)=a,, ~b(b,)=b,, ~b(~,)=c,, tk(di)=d~, c ~ ( A ) = q ~ ( B ) = c ~ ( C ) = q b ( D ) = c ~ ( E ) = I

is a retraction) and hence the embedding Lo ~ I (Lo) is pure. However, I (Lo) contains M4, the six-element lattice of height 2, as a sublattice, and L o does not, and conse- quently this embedding is not super-pure.

This obviously leaves open the following question: Is the embedding of every modular lattice into its ideal lattice pure?

REFERENCES

[1] B. Banaschewski and Evelyn Nelson, Equational compactness in equational classes of algebras, Algebra Universalis 2 (1972), 152-165.

[2] G. Birkhoff, Lattice theory, A.iV[.S. Providence, Rhode Island, 1948. [3] A. Day, Ch. Herrmann, and R. Wille, On modular lattices with four generators, Algebra Univer-

salis 2 (1972), 317-323. [4] H. Gaskill, On the relationship of a distributive lattice to its lattice of ideals, Bull. Aust. Math.

Soc. 7 (1972), 377-385. [5] I4. Gaskill, On the relationship of a distributive lattice to its lattice of ideals, Manuscript. [6] R. Wille, Jeder Endlich erzeugte, modulate verband endlicher weite ist endlich, Preprint 35, Tech-

rtische Hochschule Darmstadt. McMaster University

Hamilton, Ontario

Canada