Nearring structure of variants of some transformation semigroups

Abstract

A left [right] nearring is a system (N,+,) such that (N,+) is a group, (N,) is a semigroup and the operation left [right] distributes over the operation +. For a semigroup , 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 O adjoined. We say that a semigroup S admits a left [right] nearring structure if there exists an operation + on S[Superscript 0] such that (S[Superscript 0], +, ) is a left [right] nearring. For a semigroup S and a =S, define an operation * on S by x*y =xay for all x, y = S. The semigroup (S,*) is called a variant of S and (S,*) is denoted by (S, a). Let X be a set and P(X) be the set of all transformations from subsets of X into X. Hence P(X) is a semigroup under composition. Let V be a vector space over a division ring R and L[Subscript R (V)] be the set of all linear transformations on V. Then L[Subscript R (V)] is a semigroup under composition. Various types of subsemigroups of variants of P(X) and subsemigroups of variants of L[Subscript R (V)] are studied. Main results are determining when these semigroups admit the structure of a left [right] nearring.