Multi-valued homomorphisms of semigroups and regularity of semigroups of multi-valued functions

Abstract

An element x of a semigroup S is said to be regular if x = xyx for some y is an element of a set S,and S is called a regular semigroup if every element of S is regular.A multi-valued function f from a semigroup S into a semigroup S’ is called a multi-valued homomorphism if F(xy)=f(x)f(y)(={st I s is an element of a set f(x))and f(y)}) for all x,y is an element of a set S. For a semigroup S, let MHom(S) be the semigroup of all multi-valued homomorphisms of S under composition and let SMHom(S) be the subsemigroup MHom(S) consisting of all f is an element of Mhom( S ) satisfying the condition union [subscript x is an element of a set S]f(x) = S.Let (Z,+) and (Z[subscript n ],+ ) be the additive group of integers and the additive group of integers modulo n, respectively.Elements of MHom (Z, +), MHom (Z[subscript n ], +), SMHom(Z, +) and SMHom(Z[subscript n], +) have been already characterized. In this research, we characterize the regular elements semigroups MHom (Z, +), MHom(Z[subscript n ],+), SMHom(Z,+) and SMHom(Z[subscript n],+) and give a characterization determining when MHom (Z[subscript n ], +) and SMHom(Z[subscript n],+) are regular semigroups.We also characterize the elements of MHom(S) where S is one of the following semigroups : a left zero semigroup, a right zero semigroup, a zero semigroup and a Kronecker semigroup.In addition, some sufficient conditions for f is an element of a set MF(X) to be regular are given where MF(X) is the semigroup, under composition, of all multi-valued functions of a nonempty set X.