On a characterization of o-modular semilattices

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.