271

A NOTE ON WHITMAN'S PROPERTY FOR FREE LATTICES

R A C H A D A N T O N I U S and I V A N R I V A L

A lattice L satisfies Whi tman ' s property whenever

(W) for all a, b, c, d e L , i f a A b < c v d then a < c v d or b < c v d or a A b < c or a A b < d .

Every free lattice satisfies (W); in fact, (W) is one o f four condit ions used by

P. M. Whi tman [4] to characterize free lattices. On the other hand, the lattices L 1

and L 2 illustrated below do not satisfy (W). B. J6nsson and J. E. Kiefer [3] investigated finite lattices L which, in addit ion

to (W), satisfy a special case o f the distributive law, namely,

( S D v ) f o r all a, b, c e L , i f a v b = a v c then a v b = a v (b ^ c),

and

( S D ^ ) for all a, b, c e L , i f a A b = a A c then a ^ b = a ^ (b v c).

Using P. M. Whi tman ' s canonical representations o f elements o f a free lattice,

B. J6nsson [2] had earlier shown that every free lattice satisfies ( S D v ) and (SD^) . It is easy to verify that L~ and L 2 satisfy ( S D v ) and (SD^).

It is the purpose of this note to show that, in the presence of ( S D v ) and (SD^) , the

lattices L~ and Lz are, indeed, characteristic of (W).

T H E O R E M . Let L be a lattice with no infinite chains and which satisfies (SD v) and (SD^) . Then L satisfies (W) if and only i f it contains no sublattice isomorphic to

L 1 or L 2. Proof. In the light of the preceding remarks we need only establish the sufficiency.

Let us suppose that L contains no sublattice isomorphic to LI or L 2. Tha t L contains no sublattice isomorphic to L1 is equivalent to : every element o f

L is either join-irreducible or meet-irreducible. Now, if L fails to satisfy (W) then

there exist a, b, c, d e L such that a A b < c v d but a z g c v d, b ~ c v d, aAbegc , and a ^ b :gd. Since all chains in L are finite, a and b may be so chosen that a covers a ^ (c v d) and b covers b ^ (c v d). I f a v c v d and b v c v d both cover c v d then a v c v d = b v c v d since c v d is meet-irreducible. This, however, violates (SD v);

hence, we may assume that there exists e e L such that c v d < e < a v c v d, and, since a covers a A (C V d), a A (C V d) = a A e.

We now choose c', d ' e L such that c < c', d<_ d', c' v (a ^ e) covers c', and d ' v (a ^ e) covers d ' . As above, i fa ^ e covers both c' ^ a A e and d ' ^ a ^ e then c' A a A e = d ' ^ a A e.

Therefore, by (SD ^ ), we may assume that there existsfe L such that c ' ^ a ^ e < f < a ^ e

Presented by R. McKenzie. Received January 23, 1974. Accepted for publication in final form February 15, 1974.

272 RACHAD ANTONIUS AND IVAN RIVAL

and c' v (a A e) = c' v f . Finally, the elements c' A f , e', f , a ^ e, c ' v f , a, (a v c' q f ) A e, and a v c ' v f fo rm a sublatt ice of L i somorphic to L2, cont rary to our assumpt ion . This completes the proof.

The requirement that L have no infinite chains cannot be d ropped (cf. [ 1 ]) .

REFERENCES

L~ L 2

[1] I--I. Gaskill, G. Grfitzer, and C. R. Platt, Sharply transferable lattices, (to appear). [2] B. J6nsson, Sublattices of a free lattice, Canad. J. Math. 13 (1961), 256-264. [3] B. J6nsson and J. E. Kiefer, Finite sublattices of a free lattice, Canad. J. Math. 14 (1962), 487--497. [4] P. M. Whitman, Free lattices, Ann. Math. 42 (1941), 325-330.

University de Montreal Montreal, Quebec

Canada

University o f Manitoba Winnipeg, Manitoba Canada