Free Lattices over Halflattices

0. INTRODUCTION. Although the word problem for free lattices is well known to be solvable (cf. Dean [1]), the question still remains open to characterize the finite partial lattices P for which the free lattice F (P ) over P is finite. There are partial answers to this question. In Wille [5] the problem is solved for the partial lattices P that are both meet– and join–trivial in the sense that whenever the meet xy or the join x+ y of two elements x, y is defined in P then the elements are comparable. In [3] the problem is solved for join–trivial partial lattices. In the papers [2] and [4] free lattices over partial lattices from some other special classes are investigated. In the present paper we shall be concerned with free lattices over halflattices. By a halflattice we mean a partial lattice P such that xy is defined for all pairs x, y ∈ P and x + y is defined whenever x, y are two elements with a common upper bound in P . It is easy to see that a partial lattice P is a halflattice iff there exists a lattice L containing P as a relative sublattice such that P is an order–ideal in L (i.e., a ∈ P implies b ∈ P for all b ∈ L with b ≤ a); for a given P we can define L by L = P ∪ {1} where 1 is the greatest element of L. We shall not solve in this paper the problem for which halflattices P is the free lattice over P finite. However, we shall prove that F (L) can be finite under a very restrictive condition only. Namely, we prove that if F (P ) is finite for a finite halflattice P then the set of the elements of F (P )− P that can be expressed as x+ y for some x, y ∈ P is a chain of at most four elements. And we give an example showing that the number four is possible in this context. For the terminology and notation see our paper [3]; here we shall only briefly recall the construction of the free lattice F (P ) over a partial lattice P . The algebra of terms over P is denoted by T (P ). For every term t define an ideal ↓t and a filter ↑t of P by ↓t = {a ∈ P ; a ≤ t} and ↑t = {a ∈ P ; a ≥ t} for t ∈ P , ↓t = ↓t1 ∨ ↓t2 and ↑t = ↑t1 ∩ ↑t2 for t = t1 + t2, ↓t = ↓t1 ∩ ↓t2 and ↑t = ↑t1 ∨ ↑t2 for t = t1t2. Define a binary relation ≤ on T (P ) as follows: if u ∈ P and v ∈ T (P ) then u ≤ v iff u ∈ ↓v; if u ∈ T (P ) and v ∈ P then u ≤ v iff v ∈ ↑u; if u = u1 + u2 then u ≤ v iff u1 ≤ v and u2 ≤ v; if v = v1v2 then u ≤ v iff u ≤ v1 and u ≤ v2; if u = u1u2 and v = v1 + v2 then u ≤ v iff either u ≤ v1 or u ≤ v2 or u1 ≤ v or u2 ≤ v or u ≤ a ≤ v for an element a ∈ P . Then ≤ is a quasiordering and the relation ∼ on T (P ) defined by u ∼ v iff u ≤ v and v ≤ u is a congruence. The free lattice over P is isomorphic to T (P )/ ∼.