Finite sublattices of a free lattice

Every finite semidistributive lattice satisfying Whitman's condition is isomorphic to a sublattice of a free lattice. Introduction. The aim of this paper is to show that a finite semidistributive lattice satisfying Whitman's condition can be embedded in a free lattice. This confirms a conjecture of Bjarni Jónsson, and indeed our proof will follow the line of approach originally suggested by him in unpubhshed notes around 1960. This approach was later described in Jónsson and Nation [15], to which the reader is referred for a more complete discussion of the background material and related work than will be given here. Let us recall some relevant definitions and results. A finite sublattice of a free lattice satisfies Whitman's condition [23] (W) ab < c + d iff a < c + d or b < c + d or ab < c or ab < d and the semidistributive laws introduced by Jónsson [12] (SDV) u = a + b = a + c implies u = a + be, (SDA) u = ab = ac implies u = a(b + c). As in [15], we shall refer to a finite lattice satisfying these three conditions as an S-lattice. We will often use the following (equivalent) form of the semidistributive laws [14]. (SDV) u = 2 a,, 2 bj implies u = 2,2, a,bp (SDA) u = n a,: = LI bj implies u = II, II, (a,. + bj). Let J(L) denote the set of nonzero join-irreducible elements in a finite lattice L. Every element p G J(L) has a unique lower cover, which we will denote by p^. If />„ G J(L), letpt<1 = (pf)+. Dually, M(L) denotes the set of nonunit meet-irreducible elements of L, and for y G M(L), y* >y. In a finite semidistributive lattice there is a bijection between J(L) and M(L), p <-» k(p) = 2 {x G L: x > pt andx £p}. (In fact, A. Day has shown that this characterizes finite semidistributive lattices [4].) Now px = p, iff x > p„ and x %p, and, by (SDA), p/c(p) = p„; thus k(p) is the largest element in L with this property. Repeatedly we will use the following observations. Received by the editors November 19, 1980. 1980 Mathematics Subject Classification Primary 06B25; Secondary 08B20. 'This work was supported in part by the National Science Foundation Grant No. MCS79-01735. © 1982 American Mathematical Society 0002-9947/82/0000-0321/S07.75 311 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use