Generalized transformation semigroups admitting hyperring structure

Abstract

A semigroup S is said to admit a hyperring structure if there exists a hyperoperation + on S[superscript 0] such that (S[superscript 0], +, .) is a (Krasner) hyperring where . is the operation of S[superscript 0]. For a semigroup S and theta sigma S[superscript 1], let (S, theta) be the semigroup S under the operation * defined by x * y = x-theta-y for all x, y sigma S. The full transformation semigroup on a nonempty set X is denoted by T(X). For a vector space V over a division ring, let L(V) be the semigroup of all linear transformations alpha : V vector V under composition. In this research, we give characterizations determining when the semigroup (S, theta) with theta sigma S[superscript 1] admits a hyperring structure where S is any of the following subsemigroups of T(X) and of L(V) : T(X), M(X) = {alpha sigma T(X) | alpha is 1 - 1}, E(X) = {alpha sigma T(X) | Im-alpha = X} T[subscript 1](X) = {alpha sigma T(X) | Im-alpha is finite}, T[subscript 2](X) = {alpha sigma T(X) | X \ Im-alpha is finite}, T[subscript 3](X) = {alpha sigma T(X) | K(alpha) is finite} where K(alpha) = {x sigma X | alpha is not 1 - 1 at x}, T[subscript 4](X) = {alpha sigma T(X) | alpha is 1 - 1 and X \ Im-alpha is infinite} where X is infinite, T[subscript 5](X) = {alpha sigma T(X) | K(alpha) infinite and Im-alpha = X} where X is infinite, L(V), M(V) = {alpha sigma L(V) | alpha is 1 - 1}, E(V) = {alpha sigma L(V) | Imalpha = V} L[subscript 1](V) = {alpha sigma L(V) | dim Im-alpha is finite}, L[subscript 2](V) = {alpha sigma L(V) | dim (V / Im-alpha) is finite}, L[subscript 3](V) = {alpha sigma L(V) | dim Keralpha is finite} L[subscript 4](V) = {alpha sigma L(V) | alpha is 1 - 1 and dim (V / Im-alpha) is infinite} where V is infinite dimensional, L[subscript 5](V) = {alpha sigma L(V) | dim Ker-alpha is infinite and Im-alpha = V} where V is infinite dimensional.