On cardinalities of Green’s equivalence classes in the semigroup of difunctional binary relations
J. Ibrahim, I. M. Sawudi and A. T. Imam
Keywords: Binary relations, Difunctional relations, Green’s relations, Cardinalities
In this paper, we considered the set DX, consisting of all binary relations satisfying ( ) . This set is an inverse semigroup under a binary operation defined by xα = yβ−1 ∅, where xα denotes the set of images of x under α, and yβ−1 denotes the set of pre-images of y under β. Combinatorial results relating to Green’s relations in semigroup are obtained. In particular, we obtained cardinalities of Green’s equivalence classes in the semigroup for the case where X is finite. Also, we obtained the number of idempotent elements in to be equal to , where n = |X| and B(k) is the Bell number defined as the number of partitions of a set of k elements.