Semigroups of linear transformations with fixed subspaces: Green’s relations, ideals and finiteness conditions

Abstract

© 2019 World Scientific Publishing Company Let (Formula presented.) be a vector space and (Formula presented.) denote the semigroup (under the composition of maps) of all linear transformations from (Formula presented.) into itself. For a fixed subspace (Formula presented.) of (Formula presented.), let (Formula presented.) be the subsemigroup of (Formula presented.) consisting of all linear transformations on (Formula presented.) which fix all elements in (Formula presented.). In this paper, we describe Green’s relations, regularity and ideals of (Formula presented.); and characterize when (Formula presented.) is factorizable, unit-regular and directly finite, from which the results on (Formula presented.) can be recaptured easily when taking (Formula presented.) as a zero subspace of (Formula presented.).