Linear transformation semigroups admitting the structure of a semihyperring with zero

Abstract

A semiring is a symtem (A, +,.) such that (A, +) and (A,.) are semigroups and the operation . distributes over the operation +. A semiring (A, +, .) is additively commutative (AC) if x + y = y + x for all x, y [is an element of] A. The zero of a semiring (A, +,.) is an element 0 [is an element of] A such that x + 0 = 0 + x = x and x . 0 = 0 . x = 0 for all x [is an element of] A. For a semigroup S, let S[superscript 0] be S if S has a zero and S contains more than one element, otherwise, let S[superscript 0] be the semigroup S with a zero 0 adjoined. We say that a semigroup S admits the structure of a [AC] semiring with zero if there exists an operation + on S[superscript 0] such that (S[superscript 0], +, .) is a [AC] semiring with zero where . is the operation on S[superscript 0]. A hyperoperation on a nonempty set H is a function ํ: H x H > P*(H) where P(H) is the power set of H and P*(H) = P(H)\{[is an empty set]}. For this case, (H, ํ) is called a hypergroupoid. For a hypergroupoid (H, ํ) and nonempty subsets X and Y of H, we let XํY denote the union of all sets xํy where x and y run over X and Y, respectively. A semihypergroup is a hypergroupoid (H, ํ) with (xํy)ํz = xํ(yํz) for all x, y, z[is an element of] H. A semihyperring is a system (A, +, .) satisfying the following properties: (A ,+) is a semihypergroup, (A, .) is a semigroup and the operation . is distributive over the hyperoperation +. The zero of a semihyperring (A, +, .) is an element 0 [is an element of] A such that x + 0 = 0 + x = {x} and x . 0 = 0 . x = 0 for all x [is an element of] A. Also, a semihyperring (A, +, .) is additively commutative (AC) if x + y = y + x for all x, y [is an element of] A. Semigroups admitting the structure of a semihyperring with zero are defined analogously. Let V be a vector space over a division ring R and L[subscript R](V) the semigroup under composition of all linear transformations [alpha] : V > V. By a linear transformation semigroup on V we mean a subsemigroup of L[subscript R](V). A partial linear transformation of V is a linear transformation from a subspace of V into V. Various types of linear transformation semigroups are studied. We determine when they admit the structure of a semihyperring with zero. It is shown that semigroups without zero always admit the structure of an AC semihyperring with zero and the structure of a semiring with zero. However, we characterize when our target linear transformation semigroups without zero admit the structure of an AC semiring with zero. Moreover, the partial linear transformation semigroup on V is studied. Necessary conditions for this semigroup to admit the structure of an AC semiring with zero are given.