On the power graphs of semigroups of homogeneous elements of graded semisimple Artinian rings

Abstract

Let S be a groupoid (magma) with zero 0, and let R = ⊕ s ∈ S R s be a contracted S-graded ring, that is, an S-graded ring with R 0 = 0. By G ( H R ) we denote the undirected power graph of a multiplicative subsemigroup H R = ∪ s ∈ S R s of R, and by G * ( H R ) a graph obtained from G ( H R ) by removing 0 and its incident edges. If Re is a nonzero ring component of R, then G * ( R e ) denotes a subgraph of G * ( H R ) , induced by R e * . In this paper we address a problem raised in [Abawajy, J., Kelarev, A., Chowdhury, M.: Power Graphs: A Survey. Electron. J. Graph Theory Appl. 1(2), 125–147 (2013)]. Namely, let S be torsion-free, that is, s n = t n implies s = t for all s, t ∈ S , and all positive integers n, and let S be 0-cancellative, that is, for all s, t, u ∈ S , su = tu ≠ 0 implies s = t , and us = ut ≠ 0 implies s = t . Also, let R be semisimple Artinian. We prove that if G * ( R e ) is connected for every nonzero ring component Re of R, then the connected components of G * ( H R ) are precisely the graphs G * ( R e ) .