Abstract:
According to Clifford and Preston (1961), the theory of semigroups began with a paper by Suschkewitsch in 1928. In this work, Suschkewitsch used transformations of a finite set to describe the structure of a certain algebraic semigroup, thus as observed by Sullivan in 2000, "at the very beginning, transformation semigroups were recognised as being useful" by providing interesting examples to illustrate concepts in the theory of semigroups. Since then, important results on ideals, morphisms, congruences and generators of semigroups of transformations have been obtained, and some of these results have corresponding ones in the context of linear algebra.
As noted by Sullivan in 1991, results were first proved for transformations of a set, and then analogous statements were studied for the vector space setting. But the ideas and techniques used in these two areas are quite different, and often there are 'puzzling' differences between the results obtained. Thus, there is an ongoing interest in linear transformation semigroups and, in the last two decades, many papers have been written on this topic. Throughout this thesis, we will study several semigroups of linear transformations.
In chapter 2, we consider the semigroup GS(m;n) consisting of all one-to-one linear transformations of a vector space V?into itself whose ranges have codimension n, where N₀ £ n £ m = dim V. This is a linear version of the well-known Baer-Levi semigroup BL(p;q) defined on a set X, where N₀ £ q £ p = |X|. We show that these two types of Baer-Levi semigroup are never isomorphic, although they have the same basic properties; and we prove that the semigroups GS(m;n) act as a model for right simple, right cancellative semigroups without idempotents. Moreover, we determine the left ideals of GS(m;n) and some of its maximal subsemigroups.
In chapter 3, we study the semigroup KN(m;n) of all one-to-one linear transformations of V into itself whose ranges have codimension at least n, where N₀ £ n £ m = dim V. For n < m, we describe the Green?s relations and ideals of KN(m;n), and show that it has infinitely many bi-ideals which are not quasi-ideals. Also, we prove that KN(m;n) acts as a model for right cancellative semigroups without idempotents whose right ideals form a chain and which have no maximal principal left ideals.
The semigroups considered in chapters 2 and 3 are defined by restricting certain cardinalities: namely, the dimension of the kernels and the codimension of the ranges of their elements. Analogously, in chapter 4 we consider four semigroups of linear transformations defined by fixing an infinite upperbound or lowerbound on one of these cardinalities. We determine whether these four semigroups belong to the class of all semigroups whose sets of bi-ideals and quasi-ideals coincide, and, for each semigroup, we describe its maximal regular subsemigroup, Green's relations and ideals. We also determine all the maximal right simple subsemigroups of one of the semigroups, and, using the work by Reynolds and Sullivan in 1985 and by Sullivan in 1994, we discuss products of idempotents and products of nilpotents.
In chapter 5, we show that the symmetric inverse semigroup I(X) defined on a set X?is almost never isomorphic to the semigroup I(V) consisting of all injective partial linear transformations of a vector space V, and prove that any inverse semigroup can be embedded in some I(V). In 1999, Yang classified the maximal inverse subsemigroups of all the ideals of I(X) for finite X, and we do the same for I(V) where V is finite-dimensional. In 1984, Howie and Marques-Smith showed that, if p?= q, then BB^-1 = I(X), where B = BL(p; q) with |X| = p ³ q ³ N₀, and they described the subsemigroup of I(X) generated by B^-1B. A decade later, Lima extended that work to 'strong independence algebras', and thus also to vector spaces. Here, we answer the natural question: what happens when  p > q? We also show that, in this case, BB^-1 and its analogue for vector spaces are never isomorphic.
In chapter 6, we consider the problem of describing all isomorphisms between certain semigroups of linear transformations. We extend a result by Sullivan on the automorphisms of subsemigroups of T(X) covering X to isomorphisms of semigroups of transformations covering the underlying sets, and we compare this result with the work by Schwachhöfer and Stroppel in 1997. Although the problem of describing all isomorphisms between linear Baer-Levi semigroups is as yet unsolved, we present some results in that direction. Finally, we extend Baer's First Fundamental Theorem of Projective Geometry to certain posets of subspaces which may be important for a complete solution of this problem.