Semigroups admitting nearring structure

Abstract

A system (N, +, .) is called a right [left] nearring if (i) (N, +) is an abelian group, (ii) (N, . ) is a semigroup and (iii) (x+y) . z = x . z + y . z [z . (x +y) = z . x + z . y] for all x, y, z [is an element of a set] N A semigroup S is said to admit a right [left] nearring structure if (1) (S, +, .) is a right [left] nearring for some operation + on S where . is the operation on S or (2) (S[superscript 0], +, .) is a right [left] nearring for some operation + on S[superscript 0] where . is the operation on S[superscript 0] For a nonempty set X, let G(X), T(X), P(X) and I(X) denote respectively the symmetric group on X, the full transformation semigroup on X, the partial transformation semigroup on X and the symmetric inverse semigroup on X. In this research, we characterize when G(X), T(X), P(X) and I(X) admit a right nearring structure and a left nearring structure. We also consider the corresponding idea for certain matrix groups and some particular semigroups.