A New Local Criterion for the Lattice Property

Abstract. We prove that a bounded poset of finite length is a lattice if and only if the following condition holds: whenever two elements x 1 , x 2 that cover a common element x are both smaller that two elements y 1 , y 2 that are covered by a common element y, then there exists an element z that is an upper bound for x 1 , x 2 and a lower bound for y 1 , y 2 . For posets that arise in various combinatorial and geometric situations it is nontrivial to check whether they are lattices. In [2] it was observed that for this it suffices to check a local condition: the existence of a join for any pair of elements that cover a common element. In this note we give a new local criterion for the lattice property: it only considers two covers of an element x, which are both smaller than two cocovers of an element y, and requires that there is an element z between the two covers of x and the two cocovers of y. Furthermore, our new criterion is self-dual (that is, equivalent to its order dual formulation), and it does not any more presuppose the existence of lattice operations for any pair of elements. It was used in [4] to study the lattice property for higher Bruhat orders. For background on elementary properties of posets see [1] and [3, Chap. 3]. We need the following concepts and terminology. All posets P considered in this note are bounded: they have a unique minimal element0 and a unique maximal element1. A chain in a poset is a totally ordered subset, its length is the cardinality minus 1. The length ℓ(P ) of a poset is the maximal length of a chain in P . The posets we consider need not be finite, but they have finite length. The closed interval {z ∈ P : x ≤ z ≤ y} is denoted by [x, y]