On Cardinalities of Green’s Equivalence Classes in the Semigroup of Difunctional Binary Operation
Keywords:Quasi-idempotent, Gravity, ,Depth, Cycle, Fixed point, Orbit
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.
East, J. and Vernitski, A.: Ranks of ideals in inverse semigroups of difunctional binary relations. Semigroup Forum. 96, 21 - 30 (2018)
Jaoua, A., Elloumi, S., Hasnah, A., Jaam, J. and Nafkha, I.: Discovering regularities in databases using canonical decomposition of binary relations. Journal on Relational methods in Computer Science. 1, 217 - 234 (2004)
Kudryavtseva, G. and Maltcev, V.: Two generalizations of the symmetric inverse semigroup. Publicationes Mathematicae Debrecen 78, 253 - 282 (2011)
Riguet, J.: Relations binaire, fermetures, correspondances de galois. Bulletin Society Mathematics France. 76, 114 - 155 (1948)
Vernitski, A.: A generalization of symmetric inverse semigroups. Semigroup Forum 75, 417 - 426 (2007)