Graphs Based on BCK/BCI-Algebras

Many authors studied the graph theory in connection with commutative semigroups and commutative and noncommutative rings as we can refer to references. For example, Beck 1 associated to any commutative ring R its zero-divisor graph G R whose vertices are the zero-divisors of R including 0 , with two vertices a, b joined by an edge in case ab 0. Also, DeMeyer et al. 2 defined the zero-divisor graph of a commutative semigroup S with zero 0x 0 ∀x ∈ S . In this paper, motivated by these works, we study the associated graphs of BCK/BCIalgebras. We first introduce the notions of l-prime quasi-ideals and zero divisors and investigated related properties. We introduce the concept of associative graph of a BCK/BCIalgebra and provide several examples. We give conditions for a proper quasiideal of a BCK/BCI-algebra to be l-prime. We show that the associative graph of a BCK-algebra is a connected graph in which every nonzero vertex is adjacent to 0, but the associative graph of a BCI-algebra is not connected by providing an example.