On Cardinalities of Green’s Equivalence Classes in the Semigroup of Difunctional Binary Operation
Keywords:
Quasi-idempotent, Gravity, ,Depth, Cycle, Fixed point, OrbitAbstract
In this paper, we considered the set DX, consisting of all binary relations α ⊆ X × X satisfying (∀x,y,u,v ∈ X) (x,u),(x,v),(y,u)∈α ⇒(y,v) ∈ α. 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.
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