On a characterization of o-modular semilattices

TL;DR

This paper explores a characterization of o-modular semilattices, simplifying existing proofs and clarifying the conditions under which the characterization holds, especially for complete algebras.

Contribution

It provides a simplified proof of a characterization of o-modular semilattices and clarifies the conditions for the biconditional characterization to hold.

Findings

Simplified the proof of the characterization of o-modular semilattices.

Identified that the proof holds only for complete algebras under the biconditional.

Clarified the conditions necessary for the characterization to be valid.

Abstract

Dedekind stated and proved the well-known fact that a lattice is modular if and only if it does not contain a pentagon as a sublattice. In this paper we consider a similar result in the literature for the case of certain class of modular join semilattices. We both simplify the original proof of the mentioned result and note that if the notion of characterization is understood strictly as a biconditional, then the proof only holds for the complete algebras in the class.