On Some Studies Of Linear Transformation Semigroups

In this dissertation, we study some properties of several special linear transformation semigroups. The main results are given in follow .The Chapter 1 is the section of introduction .The Chapter 2 gives the characterizations of a nonzero principle quasi-ideal to be a 0-minimal quasi-ideal of the quasi-linear transformation semigroups (QL_H(V, W, k),θ) and QL_H(V) respectively;And also gives characterizations of a 0-minimal quasi-ideal to be a 0-minimal ideal of the semigroups (QL_H(V, W, k),θ). The main results are given as following :Theorem 2.1.15 For α ∈ QL_H(V, W, k)\{0}, (α)_q is a 0-minimal quasi-ideal of (QL_H(V, W, k), θ) if and only if one of the following three conditions holds:(1) rank α = 1, im α (?) ker θ and im θ (?) ker α;(2) im α (?) ker θ;(3) im θ (?) ker α.Theorem 2.2.3 For α ∈ QL_H(V)\{0}, the following statements are equivalent in QL_H(V):(1) (α)_q is a 0-minimal quasi-ideal of QL_H(V);(2) rank α= 1;(3) (α)_Q = Hα.Theorem 2.3.1 In a nonzero semigroup (QL_H(V, W, k), θ), every 0-minimal quasi-ideal is a 0-minimal ideal if and only if θ = 0 or dim V =dim W = 1.The Chapter 3 mainly discusses the regularity of the generalized linear transformation semigroup (L_D(V, W, k), θ).And also gives a characterization of the regularity of this semigroup. The main result is given as following:Theorem 3.2 If there is α ∈ L_D(V, W, k)\{0}, such that im α (?) ker θ or im θ (?) ker α, then the semigroup (L_D(V, W, k), θ) is not regular.Theorem 3.3 For a G Ld{V, W, k), such that im a $￡ker 9 and im 9 <￡. ker a. If rank a — 1, then the semigroup {Ld{V, W, k), 9) is regular.Theorem 3.6 The semigroup {Ld(V, W, k), 9) is regular if and only if V — {0}, W — {0} or 9 is an isomorphism from W onto V.The chapter 4 discusses the simplity or regularity of some subsemigroups of the linear transformation semigroup Ld(V).The main results'are given as following:For any dimension of the vector space V.Theorem 4.5 The subsemigroup AM(V) is regular .Theorem 4.6 The subsemigroup AE(V) is regular .The chapter 5 mainly studies the Green's relations in the semigroup L(V, p, W) and characterizes the regular elements of the semigroup L(V, p, W). The main results are given as following:Theorem 5.1.5 For a, /3 6 L(V, p, W), then aH/3 if and only if Ja = W/3.Theorem 5.1.7 For a,p 6 L{V, p, W), then a￡p if and only if ▼? ?-? T/3.Theorem 5.1.8 For a, (3 € L(V, p, W), then aP/3 if and only if there is an isomorphism linear transformation $ : im a -> im ft such that (J^nim a)$ CW and (Ta)$ 4-> ▼/?.Theorem 5.2.1 For a G L(V, p, W), then a is regular if and only if im a/p C Tq.