Characterizations of connected orthogonality graphs of projections of Rickert *-rings

Abstract

In this paper, we study the orthogonality graphs (see Definition 1.2) of ortholattices. We provide a graph theoretic condition for an ortholattice to be orthomodular. We prove that, the orthogonality graphs of two orthomodular lattices are isomorphic if and only if the lattices are isomorphic. As an application, it is proved that the zero-divisor graph of a Rickart ∗-ring is obtained by successively duplicating the vertices of the orthogonality graph of the lattice of projections in the ring. We characterize the finite Rickart ∗-rings for which the orthogonality graph of projections is connected.

Mathematics Subject Classification (2010): Primary 05C38, Secondary 05C12.

Key words: Orthogonality graph, ortholattice, orthomodular lattice, zero-divisor graph.