Zero-divisor graphs of lower dismantlable lattices-II

TL;DR

This paper investigates the zero-divisor graphs of lower dismantlable lattices, proving that isomorphic graphs imply isomorphic lattices under certain conditions, thus advancing understanding of lattice graph isomorphisms.

Contribution

It establishes a necessary and sufficient condition for the isomorphism of zero-divisor graphs to imply lattice isomorphism in a specific class of lattices.

Findings

Zero-divisor graphs of certain lattices are isomorphic iff the lattices are isomorphic.

Focus on lower dismantlable lattices with the greatest element join-reducible.

Provides a characterization of graph isomorphism in the context of lattice structure.

Abstract

In this paper, we continue our study of the zero-divisor graphs of lower dismantlable lattices that was started in [20]. The present paper mainly deals with an Isomorphism Problem for the zero-divisor graphs of lattices. In fact, we prove that the zero-divisor graphs of lower dismantlable lattices with the greatest element 1 as join-reducible are isomorphic if and only if the lattices are isomorphic.