On Commuting Graphs Associated to BCI-Algebras

Abstract

In this paper, first, the graph Γ(X) associated to a BCI-algebra X is studied and some related properties are established. Especially, a necessary and sufficient condition for Γ(X) to be a complete graph is given. After that, the commuting graph associated to a BCI-algebra X, denoted by G(X), is defined and some related properties are investigated. The paper provides a necessary and sufficient condition for the p-semisimple part of X to be an ideal. Moreover, a condition for an element of a BCI-algebra X to be minimal is given. Finally, it is proved that a BCI-algebra X is p-semisimple if and only if G(X) is a complete graph.

Key words and phrases. BCI-algebra, commutative BCK-algebra, ideal, p-semisimple, graph associated to a BCI-algebra, commuting graph.

2000 Mathematics Subject Classification. 06F35, 03G25, 68R10