The natural partial order on linear transformation semigroups with restrictions on nullity or co-rank

Abstract

The natural partial order ≤ on a semigroup S is a partial order de ned by a ≤ b if and only if a = xb = by and a = ay for some x; y ϵ S¹ where S¹ is the semigroup obtained from S such that S¹ = S if S has an identity and if S has no identity, let S1 be S with the identity 1 adjoined. It is known that the natural partial orders on a semigroup and its regular subsemigroups coincide. Therefore, the study of the natural partial order on nonregular semigroups are of interest. In this thesis, we give necessary and su cient conditions for elements in nonregular linear transformation semigroups with restrictions on nullity or co-rank are related under the natural partial order. Furthermore, we provide necessary and su cient conditions for elements in those linear transformation semigroups to be left and right compatible elements, minimal and maximal elements, lower and upper covers.