Studies On Order-increasing(Decreasing) Transformation Semigroups

In this dissertation. we study some properties of several special transformation semigroups. The main results are given in follow.The Section 1 mainly characterizes the starred Green's relations for the Epreserving and order increasing parital transformation semigroup .The main results are given as following:Theorem 1.6 Let f,g∈p_E⁺(X),the following statements are equivalent:i) (f,g)∈L^*:ii)Ï€(f)=Ï€(g),E(f)=E(g):iii) there exists an E^* -preserving bijectionφ:imf→img.such that g=φf.Theorem1.7 let f,g∈P_E⁺(X).the following statements are equivalent:i) (f,g)∈R^*:ii) For each A∈X/E, there exist B.C∈X/E such thatf(A∩ndomf)(?)g(B∩domg),g(A∩domg)(?)f(C∩domf):iii) there exist an E^* -admissible bijectionφ:Ï€(f)→π(g). such thatf_*=g_*φ.Theorem 1.8 let f,g∈p_E⁺(X).the following statements are equivalent:i) (f．g)∈H^*ï¼›ii)Ï€(f)=Ï€(g), E(f)=E(g), and for each A∈X/E,there exist B, C∈X/E such thatf(A∩domf)(?)g(B∩domg)．g(A∩domg)(?)f(C∩domf)ï¼›iii) There exist an E^* -admissible bijectionφ:Ï€(f)→π(g), and a E^*-preserving bijection (?):imf→img,such that f_*=g_*φ,g=(?)f．Theorem 1．11 let(f,g)∈P_E⁺(X),the following statements are equivalent:i) (f,g)∈D^*:ii) there exist an E^*-admissible bijectionφ:Ï€(f)→π(g),φis multicationof two bijections,and there exist E^*-preserving bijection (?):imf→img,such that(?)f_x=g_*φ．Corouary1．13 In P_E⁺(X),D^*=J^*．The Section 2 mainly studies the starred Green's relations for the variant of E-preserving and orde-increasing full transformation semigroups．The main resultsare given as following:Theorem 2．1 let f,g∈T_E⁺(X；θ),f≠g,the following statements areequivalent:i) (f,g)∈L^*;ii)Ï€(θf)=Ï€(f)=Ï€(g)=Ï€(θg),E(θf)=E(f)=E(g)=E(θg):iii) There exist an E^*-preserving bijectionφ:f(x)→g(x)．such that g=φf,bothθ|_f(x)andθ|_g(x)are E^*-preserving injections．Theorem2．3 let f．g∈T_E⁺(X;θ),the following statements are equivalent:i) (f,g)∈R^*;ii) For each E class A,there exist B．C∈X/E,such thatf(A)(?)gθ(B),g(A)(?)fθ(C)ï¼›iii) There exist an E_θ^*-admissible bijection (?):Ï€(f)→π(g) such thatf_*=g_*(?)．Theorem 2．4 let f,g∈T_E⁺(X;θ),the following statements are equivalent:i) (f,g)∈H^*;ii)Ï€(θf)=Ï€(f)=Ï€(g)=Ï€(θg),E(θf)=E(f)=E(g)=E(θg),while foreach E class A,there exist B,C∈X/E,such that f(A)(?)gθ(B),g(A)(?) fθ(C)Theorem 2．5 let f,g∈T_E⁺(X；θ),the following statements are equivalent:i) (f,g)∈D^*ï¼› ii) There exist an E_θ^*-admissible mapping (?):Ï€(f)→π(g) and an E^* -preserving bijectionφ:f(x)→g(x) andφcan be written as a product of two bijections.such thatφf_*=g_*(?),and bothθ|_f(x) andθ|_g(x) are E^* -preserving injections.Corollary 2.7 In T_E⁺(X;θ).there exist D^*=J^*.The Section 3 discusses the isomorphism theorom of the order-decreasing partial transformations. The main results are given as following:Theorem 3.7 let P^-(X), P^-(Y) is defined as before, the following statements are equivalent:i) X and Y are order -isomorphic:ii) p^-(X)≌P^-(Y).corollary3.8 let P^-(X), P⁺(Y) are defined as before. the following statements are equivalent:i) X and Y are order-anti-isomorphic:ii) P^-(X)≌P⁺(Y).The Section 4 mainly characteries the semigroups which satisies reg(T)= Reg(T)(T is the left ideal,right ideal, and the insection of the left ideal and right ideal of S) .The main results are given as following:Theorem 4.4 let (?) stands for the set of all left ideals of S,the following statements are equivalent:Theorem 4.5 let (?) stands for the set of all right ideals of S , the following statements are equivalent: Theorem4.6 let A stands for the set of all left ideals of S , (?) stands for the set of all right ideals of S , the following statements are equivalent:...