Co-annihilating graphs of partial transformation semigroups: structure and basic invariants
Chollawat Pookpıenlert, Nuttawoot Nupo, Yanisa Chaiya · Arab Journal of Basic and Applied Sciences · 2026
Let n∈N and Xn={1,2,…,n}. Denote by Pn the partial transformation semigroup on Xn. Clearly, the empty set ∅ is a zero element of Pn, which is written as 0. Let Pn∗=Pn\{0}. An element α∈Pn is called a zero-divisor of Pn if there exists β∈Pn∗ in which αβ=0=βα. The annihilator Ann(α) of α∈Pn is defined to be the set {β∈Pn:αβ=0=βα}. Obviously, Ann(α) always contains 0. Further, denote by Z(Pn) the set of all zero-divisors of Pn and Z(Pn)∗=Z(Pn)\{0}. In this paper, the co-annihilating graph Γ¯n of Pn is constructed as an undirected simple graph with vertex set Z(Pn)∗ and two distinct vertices α and β are joined by edge if and only if Ann(α)∩Ann(β)={0}. Certain properties of Γ¯n are investigated. Moreover, some basic invariants of Γ¯n are also determined.