9 - Congruence Semimodularity

Summary

Semilattices

Summary. Algebras whose congruence lattices satisfy nontrivial lattice identities have been thoroughly studied during the last two decades. In particular, varieties have been investigated in which the congruence lattice of all algebras is distributive or modular. Although weaker than modularity, semimodularity is still an interesting property in this connection. Ore [1942] proved that sets have semimodular congruence lattices (in fact, these are matroid lattices). Here we describe some properties of congruence lattices of semilattices that were shown to be semimodular by Hall [1971].

An algebra S = (S: ·) with a binary operation · is called a semigroup if this operation is associative, that is, x · (y · z) = (x · y) · z holds for all x, y, z ∈ S. For simplification we briefly write ab for a · b. We shall identify S with S if there is no danger of confusion. A semilattice is a semigroup satisfying xy = yx (commutativity) and x² = x (idempotency) for all x, y ∈ S. We may impose a partial order on S by defining x ≤ y if xy = x. Under this ordering, any two elements x, y ∈ S have a greatest lower bound, namely their product xy. Hence S is a (meet) semilattice.

An algebra A is said to be locally finite if every subalgebra of A generated by finitely many elements is finite. A variety ∨ is said to be locally finite if every algebra in ∨ is locally finite, or equivalently, every finitely generated algebra in ∨ is finite. Semilattices form a locally finite variety. We denote this variety by Semilattices.