Studies On Ordered Transformation Semigroup

In this dissertation, we first study the BQ property in some Ordered Transformation Semigroups , then we study the regularity in Linear Ordered Transformation Semigroup , besides we discuss the relation among 0-Minimal ideal principal ideal and 0-Minimal Quasi-ideal in a ciass of Linear Ordered Transformation Semigroup. Finally we study a class of ordered commutative semigoup and discuss the BQ property in Partial Transformation Semigroup. The main results are given in follow.In Chapter 1, we give the introduction and preliminaries.In Chapter 2, we firstly give the definition of BQ property in Ordered Transformation Semigroup;besides we find some some subsemigroups with BQ prop-erty.The main results are given in follow.Proposition 2.1.3 For any nonempty subset A of a ordered semigroup S ,(1)(S¹ A ∩ AS¹] (?) (A)_q (?) (S¹A] ∩ (AS¹];(2) (A)_b = (AS¹ A ∪A] = (ASA ∪ A ∪ A²] .Theorem 2.1.12 Let X be countable and full order ,with a minimum x₁ . If OT₁(X) = {α∈ OT{X)| α is 1-1 and X\ ran α is finite } (?0 OR(X), then(1)OT₁(X)is an ordered subsemigroup of OT(X);(2)OT₁(X) ∈ BQ if and only if X is finite .Theorem 2.1.13 Let X be countable and full order, and there exists mapping with monotone order-preserving between any two equivalent sunsets of X , and exists x₀ ∈ X such that A = {x∈ X|x ≤ x₀}, B = {x ∈ X|x > x₀} are equivalent sets with X , if OT₂(X) = {α ∈ OT(X)| α is 1-1, X\ ran a is infinite }(?) OR(X). then OT2(X) (?) BQ .In chapter 3, we study the regularity in Linear Ordered Transformation Semigroup. The main results are given in follow.Proposition 3.1 Let 9 be an order-preserving linear isomorph from Wto V, we define mapping ip : (OLD(V, W), 9) h> OLd{V) byaxj) — adfor Va € OLD(V, W);then $ is an isomorph fvom(OLD(V, W),9) to OLD(V), so {OLD{V,W),6) ^OLD{V) .Theorem 3.2 Ordered semigroup (OLp(V, W),9) is regular if and only if V = {0} or W = {0} or 0 is an an order-preserving linear isomorph from W^to V.In Chapter 4,we think of 0-Minimal Quasi-ideal in a class of Linear Ordered Transformation Semigroup. The main results are given in follow .Theorem 4.9 If a € (OLF(V, W, K),9)\{Q} , ran a ï¿¡ ker 8 and ran 9 % kera, then the following two results are equivalent :(1) (a), is a 0-minimal quasi-ideal of {OLF{V,WtK),6)\(2) rank a — 1 .Theorem 4.11 Let 9 / 0, in a nonezero ordered transformation semigroup (OLp(V, W, K), 9), every 0-minimal quasi-ideal is a 0-minimal ideal if dim V =dim 1^ = 1.Theorem 4.13 Let 9^0, every nonezero ordered transformation semigroup (OLp[V, W, K), 9) contains a 0-minimal quasi-ideal.In Chapter 5, we give some properties of regular commutative ordered semigroups relative to their prime ideal structure and also give some relations among the noetherianity, archimedeanity, regularity and the finitely generated property in the class of commutative ordered semigroups which are unions of a finite number of principal ideals.The main results are given in follow.Theorem 5.5 Let S be a regular commutative ordered semigroup and H be the collection of all ideals of S, which are not principal ( finitely generated) ideal . If H ^ 0, then there exists a prime ideal of S in H.Corollary 5.11 Let ordered semigroup S be archimedean and regular withnS = {JfaS1]. Suppose a ï¿¡ (r^-aS1] for all a e S, which is not in (AiA2 â–  â–  ? An], where Ai={x7ix G 5 | r, = 0,1, 2, â–  â–  ? }. Then S is finitely generated.In Chapter 6, we study the BQ property in Partial Transformation Semigroup, and get some subsemigroups with BQ property in Partial Transformation Semigroup. The main results are given in follow.Theorem 6.8 Ps(X) = {a 6 P{X)\ X\ ran a is finite } is a subsemigroup of P{X) , and P3(X) e BQ if and only if X is finite .Theorem 6.9 P*{X) = {a e P{X)\ a is 1-1, X\ ran a is finite } is a subsemigroup of P(X), and Pa(X) 6 BQ if and only if X is finite .