Regularities And Green's Relations For Some Transformation Semigroups

In this dissertation, we study the regularities and green's relations for some transformation semigroups. There are three chapters.the main results are given in follow.In Chapter 1, let PEOPx denote the finite E-O-preserving partial transformation semigroup on a non-empty finite totally ordered set X, we mainly study the regular elements,green's relations, and also the Green's relations for regular elements on PEOP_X ,and discuss the *-Green's relations on it .The main results are given as following:Theorem 1.1.1 Letα∈PEOP_X, thenαis a regular element if and only if for each A∈X/E,there exists B∈X/E such that imα∩A(?)(B∩domα)α.Proposition 1.1.5 The following conditions are equivalent in the semigroup PEOP_X:(1) PEOP_X is a regular semigroup;(2) R(PEOP_X) is the regular subsemigroup of PEOP_X;(3) PEOP_X = P_X or PEOP_X = POP_X;(4) E = X×X or E = 1_X.Theorem 1.2.4 Letα,β∈PEOP_X, then the following conditions are equivalent:(1) (α,β)∈L;(2) imα= imβand for each A∈X/E,there exist B, C∈X/E such that (A∩domα)α(?)(B∩domβ)β, ((A∩domβ)β(?)(C∩domα)α;(3) there exists an E~*-admissible bijection (?):π(α)→π(β) such thatα_*=(?)β_* Theorem 1.2.5 Letα,β∈PEOP_X, then (α,β)∈R if and only if there exists an E~*- order-preserving bisection (?) : imα→imβsuch thatβ=α(?).Theorem 1.2.6 Letα,β∈PEOP_X, then the following statements are equivalent:(1)(α,β)∈H;(2)imα=imβand for each A∈X/E,there exist B,C∈X/E such that (A∩domα)α(?)(B∩domβ)β, (A∩domβ)β(?)(C∩domα)α; there exists an E~*-order-preserving bijection (?):imα→imβsuch thatβ=α(?);(3)there exists an E~*-admissible bijection (?):π(α)→π(β) such thatα_* = (?)β_* and there exists an E~*- order-preserving bijection (?) :imα→imβsuch thatβ=α(?).Theorem 1.2.7 Letα,β∈PEOP_X, then (α,β)∈D if and only if there exists an E~*-admissible bijection (?) :π(α)→π(β) and an E~*-order-preserving bijection (?) : imα→imβsuch thatα_*(?)=(?)β_*.Theorem 1.3.1 Letα,β∈R(PEOP_X), then (α,β)∈L if and only if imα=imβ.Theorem 1.3.2 Letα,β∈R(PEOP_X), then (α,β)∈R if and only ifπ(α)=π(β).Theorem 1.3.3 Letα,β∈R(PEOP_X), then (α,β)∈H if and only if imα=imβandπ(α)=π(β).Theorem 1.3.4 Letα,β∈R(PEOP_X), then (α,β)∈D if and only if there exists an E~*-order-preserving bijection (?) : imα→imβ.Theorem 1.3.5 Letα,β∈R(PEOP_X), then (α,β)∈(?) if and only if there exists an E~*- order-preserving bijection (?) : imα→imβ.Theorem 1.4.4 Letα,β∈PEOP_X, if imα=imβ,then (α,β)∈L~*.Theorem 1.4.5 Letα,β∈PEOP_X, if kerα= kerβ, then (α,β)∈R~*.In Chapter 2, let PEOT_X(θ) denote the variant semigroups of finite E-Opreservingtransformation semigroup on a non-empty finite totally ordered set X, we mainly characterize the Green's relations on the PEOT_X(θ) and the Green's relations of regular elements on it. The main results are given as following: Theorem 2.1.2 Letα,β∈PEOT_X(θ), then the following statements are equivalent:(1)(α,β)∈L;(2) Xα= Xβand for all A∈X/E, there exist B, C∈X/E such that A_α(?) Bθβ, Aβ(?)Cθα;(3) there exixts an E_θ~*- admissible bijection (?) : Xα→Xβsuch thatα_*=(?)β_*.Theorem 2.1.3 Letα,β∈PEOT_X(θ), if (α,β)∈R, then there exists an E~*- preserving bijection (?) :Xα→Xβsuch thatβ=α(?).Theorem 2.2.2 Letα,β∈R(PEOT_X(θ)),then (α,β)∈L if and only if Xα= Xβ.Theorem 2.2.3 Letα,β∈R(PEOT_X(θ)), if (α,β)∈R thenπ(α)=π(β).In Chapter 3. let POP_E(X) denote the partial transformation semigroup preserving an order and an equivalent, we mainly characterize the regular elements and the Green's relations on it. The main results are given as following:Theorem 3.1.3 Let f,g∈POP_E(X), then the following conditions are equivalent:(1)(f,g)∈L;(2)π(f)=π(g),E(f)=E(g);(3) there exixts an E~*-preserving order isomorphism (?) :imf→img such that g = (?)f.Theorem 3.1.4 Let f,g∈POP_E(X), then the following conditions are equivalent:(1)(f,g)∈R;(2) imf = img, and for all A∈X/E, there exixt B,C∈X/E such that f(A∩domf)(?)g(B∩domg), g(A∩domg)(?)f(C∩domf);(3) there exixts an E~*- admissible order isomorphism (?) :π(f)→π(g)such that f_*=g_*(?).Theorem 3.1.6 Let f,g∈POP_E(X), then the following conditions are equivalent: (1)(f,g)∈D;(2) there exixt an E~*- preserving order isomorphism (?) : imf→img and an E~*-admissible order isomorphism (?):π(f)→π(g) such that g_*(?)=(?)f_*.Theorem 3.2.1 Let f∈POP_E(X), then f is regular if and only if for all A∈X/E,either A∩imf = (?), or there exists B∈X/E such that A∩imf = f(B∩domf).Corollary 3.3.3 Let f,g∈R(POP_E(X))?then the following statements hold:(1)(f,g)∈L if and only ifπ(f)=π(g);(2)(f,g)∈R if and only if imf=img;(3)(f,g)∈H if and only ifπ(f)=π(g) and imf = img.Theorem 3.3.4 Let f,g∈R(POP_E(X)), then (f,g)∈D if and only if there exists an E~* -preserving order isomorphism (?): imf→img...