Year: 2024
Type: article
Abstract: Let Xn={1,2,…,n} and Pn be a partial transformation semigroup on Xn. Obviously, the empty set ∅ is a zero element of Pn and denoted by 0. Let Pn*=Pn\{0}. An element α∈Pn is a zero divisor of Pn if there exists β∈Pn* such that αβ=0=βα. The set Ann(α)={β∈Pn:αβ=0=βα} is called the annihilator of α in Pn. It is clear that 0∈Ann(α) for all α∈Pn. Let Z(Pn) be the set of all zero divisors of Pn and Z(Pn)*=Z(Pn)\{0}. Generally, if α∈Z(Pn), then there exists β∈Z(Pn)* in which β∈Ann(α). In this paper, we construct the annihilator graph Γn of Pn which is an undirected simple graph with vertex set Z(Pn)* and two distinct vertices α and β are adjacent if and only if Ann(α)∩Ann(β)≠{0}. Furthermore, we prove some basic structural properties of Γn and determine invariants for Γn such as the diameter, girth, clique, domination number, and independence number.
Language: English
Topic: Rings, Modules, and Algebras
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: gold
Estimated APC paid: $944
Funder: Thammasat University
OpenAlex ID: W4404116786