Finiteness Conditions for Lattices

Abstract

Let ≤ denote a partial ordering on a nonempty set L and let < be defined by: a < b with a,b ∈ L if and only if a ≤ b and a ≠ b. If a,b ∈ L and b ≤ a then a is said to contain b. If M is a subset of L, then an a ∈ L such that x ≤ a, resp. a ≤ x, for all x ∈ M is said to be an upper, resp. lower, bound for the set M. An element a ∈ L is said to be the supremum, resp. infemum, of M if a is the least upperbound, resp. the largest lower bound, for M (i.e. a is an upper (resp. lower) bound for M and if a’ is another upper (resp. lower) bound for M then we have a ≤ a’, resp. a’ ≤ a). A supremum, resp. infemum of M is unique and we will denote it by sup(M) or ⋁ _x∈M x, resp. inf(M) or ⋀ _x∈M x. When using the notation ⋁ _x∈M x, resp. ⋀ _x∈M x, we will also refer to these elements as the join, resp. meet, of the elements of M. In case M = {x₁,..., x _n }, we also write sup(M) = ⋁ ⁿ_i=1 x _i = x₁ ⋁...⋁x _n and inf(M) = ⋀ ⁿ_i=1 x _i = x₁ ⋀ x₂ ⋀...x _n .