Modular and distributive semilattices

Abstract:

A modular semilattice is a semilattice $S$ in which $w \geq$ implies that there exist $x,y \in S$ such that $x \geq a,y \geq b$ and $x \wedge y = x \wedge w$. This is equivalent to modularity in a lattice and in the semilattice of ideals of the semilattice, and the condition implies the Kurosh-Ore replacement property for irreducible elements in a semilattice. The main results provide extensions of the classical characterizations of modular and distributive lattices by their sublattices: A semilattice $S$ is modular if and only if each pair of elements of $S$ has an upper bound in $S$ and there is no retract of $S$ isomorphic to the nonmodular five lattice. A semilattice is distributive if and only if it is modular and has no retract isomorphic to the nondistributive five lattice.