Some regular equivalence relation on the semihypergroup of the partial transformation semigroup on a set and local subsemihypergroups with that regular equivalence relation

Abstract

A hyperoperation ○ on a nonempty set H is a function from H x H into P∗(H) where P∗(H) is the set of all nonempty subset of H and (H, ○) is call a hypergroupoid. A hypergroupoid (H, ○) is called a semihypergroup if the hyperoperation ○ is associative. Thus, semihypergroups generalize semigroups. Moreover, if S is a semigroup; we can define a hyperoperation ○ on S in order to make (S, ○) a semihypergroup. In 2013, R.I. Sararnrakskul defined a hyperoperation ○ on the partial transformation semigroup P (X) to make a semihypergroup. In this paper, we define a regular equivalence relation ρon (P (X), ○) so that P (X)/ρ is a semihypergroup and then we studies some subsemihypergroup of P (X)/ρ. © 2015 Academic Publications, Ltd.