The Graphs Associated with Ordered Structures and Algebraic Structures

Abstract

This project focuses on the interplay between the zero-divisor graphs of semigroups and zero-divisor graphs of meet-semilattices. Mainly, we examine the weakly perfectness of the zero-divisor graphs of semigroups and annihilating-ideal graphs of semigroups. In particular, we solve DeMeyer and Schneider (L. DeMeyer and A. Schneider, The annihilating-ideal graph of commutative semigroups, J. Algebra 469 (2017), 402-420.) conjecture about the annihilating-ideal graphs of semigroups negatively. In the First chapter, we provide a new proof of an analogue of Beck's Conjecture for the zero-divisor graphs of posets. Further, we study the partial order given by LaGrange and Roy for reduced commutative semigroups. In fact, we prove that the minimal prime ideals of reduced commutative semigroups S are nothing but the minimal prime semi-ideals of S treated as a poset (under the partial order given by LaGrange and Roy). In fact, we also observe that a similar result holds for reduced commutative rings with unity. This gives a new insight about the Beck's conjecture for reduced rings via ordered sets. It is known that the set of ideals of semigroups forms a multiplicative lattice. Hence in the last section, we deal with the annihilating-ideal graphs of semigoups and its connections with the zero-divisor graphs of multiplicative lattices.