The identity-summand graph of commutative semirings

Abstract

An element r of a commutative semiring R with identity is said to be identity-summand if there exists 1 ≠ a ∈ R such that r+a = 1. In this paper, we introduce and investigate the identity-summand graph of R, denoted by Γ(R). It is the (undirected) graph whose vertices are the non-identity identity-summands of R with two distinct vertices joint by an edge when the sum of the vertices is 1. The basic properties and possible structures of the graph Γ(R) are studied. © 2014 The Korean Mathematical Society.