Several Kinds Of Semigroups And Their Properties

In this thesis,we introduce several kinds of semigroups and study some properties of them.The main results are given in following.In Chapter 1,we give the introduction and preliminaries.In Chapter 2,we introduce a new kind of subsemigroups of full transformayion semigroup T_x,that is E-order-preserving transformation semigroup EOPx.When X is finite set,we consider the regularity and Green's relations for EOP_n firstly,then consider the rank of EOP_n when the cardinal of each E-classes are equal.The main results are given in following.Theorem 2.0(1)E = X_×X_n(?)EOP_n=O_n;(2)E = I_xn(the identity relation on X_n)(?)EOP_n = T_n.Theorem 2.1.1.1 For anyα,β∈EOP_n,the following are equivalent(1)(α,β)∈(?);(2)X_nα= X_nβand(?)A∈X_n/E,(?)B,C∈X_n/E,(?)Aα(?)Bβ,Aβ(?)Cα;(3)(?);Ï€(α)→π(β)is an E^*-admissible mapping andα_* =Φβ_*.Theorem 2.1.1.2 Forα,β∈EOP_n,(α,β)∈R(?)β=αφfor some E^*-order-preserving bijectionφ;X_nα→X_nβ.Theorem 2.1.1.3 Forα,β∈EOP_n,(α,β)∈D(?)α_*ψ=ψβ_* for some E^*-admissible mappingψ;Ï€(α)→π(β)and some E^*-order-preserving bijectionΨ;X_nα→X_nβ.Theorem 2.1.2.1 Forα∈EOP_n,α∈R(EOP_n)(?)A∈X_n/E,there exists B∈X_n/E such that X_nα∩A(?)Bα.Proposition 2.1.2.4 The following are equivalent in EOP_n;(1)EOP_n is regular semigroup;(2)R(EOP_n)is a regular subsemigroup of EOP_n;(3)EOP_n = T_n or EOP_n = O_n;(4)E=X_n×X_n or E = Ix_n.Theorem 2.1.3.1 Forα,β∈R(EOP_n),(α,β)∈(?)X_nα= X_nβ.Theorem 2.1.3.2 Forα,β∈R(EOP_n),(α,β)∈(?)Ï€(α)=Ï€(β).Theorem 2.1.3.3 Forα,β∈R(EOP_n),(α,β)∈D(?)There exists an E^*-order-preserving bijectionφfrom X_nαinto X_nβ. Theorem 2.2.5.2 EOP_x=ξ₁₁^*,ξ₁₂^*,...,ξ_1(2n-2)^*,σ,ρ,π＞.Consquently,rank(EOP_x)≤2n+1.In Chapter 3,we study the sandwich semigroup T(X,Y;θ).For finite sandwich semigroup T(X,Y;θ),we consider the regularity and Greenls relations for it,study some numeral prop-erties of R(X,Y;θ),and discuss the idempotent-generated properties and the idempotent rank of RST(X,Y;θ).The main results are given in following.Theorem 3.1.1.1 Let T(X;θ)be the variant semigroup on Tx,then(1)T(X,X;θ)= T(X;θ)when X = Y.(2)T(X,X;θ)= Tx when X;Y andθ= 1|x.Theorem 3.1.1.2 Let MO = {aθ;α∈T(X,Y;θ)},then(2)Mθ(?)T(X,Y;θ)whenθis injective.(3)Mθ= Tx and T(X,Y;θ)(?)Tx whenθis bijective.Theorem 3.1.2.1 For anyα,β∈T(X,Y;θ),α≠β.Then(α,β)∈(?)Xα= Yθα= Yθβ= Xβ.Theorem 3.1.2.2 For anyα,β∈T(X,Y;θ),α≠β.Then(α,β)∈(?)ker(α)= ker(β),and bothθ|xαandθ|xβare injective.Theorem 3.1.2.4 Letα=(?)∈T(X,Y;θ),then(2)α∈R(T(X,Y;θ))(?)θ|xαis injective and A_i∩Yθ≠φfor i = 1,2,...,γ.(?)|L_α≥2 and |R_α|≥2,or |L_α| = 1 and |R_α| =|Y|.Theorem 3.1.2.5 For anyα,β∈T(X,Y;θ),α≠β.Then(α,β)∈D if and only if the following.(a)L_α= L_βand bothθ|X_αandθ|X_βare not injective.(b)R_α= Rβand |X_α|= |Xβ|＞|Yθα| = |Yθβ|.(c)Bothθ|X_αandθ|Xβare injective,and |X_α| = |Yθα| = |Yθβ| = |Xβ|.Theorem 3.1.3.1 R(T(X,Y;θ))is a regular subsemigroup of T(X,Y;θ).Theorem 3.1.3.4 For anyα,β∈R(T(X,Y;θ)),α≠β.Then(α,β)∈(?)^R(?)X_α= Xβ. Theorem 3.1.3.5 For anyα,β∈R(T(X,Y;θ)),α≠β.Then(α,β)∈R^R(?)ker(α)= ker(β).Theorem 3.1.3.6 For anyα,β∈R(T(X,Y;θ)),α≠β.Then(α,β)∈D^R(?)|Xα|= |Xβ|.Theorem 3.2.3.5 RST(X,Y;θ)=＜E(D_r-1)＞.Theorem 3.4.2 For any I(?)E(D_r-1),RST(X,Y;θ)=＜I＞(?)Γ(I)is strongly connected R-complete graph, whereΓ(I)is the directed graph accompany with I.Theorem 3.4.7 Let irank(RST(X,Y;θ))be the idempotent rank of RST(X,Y;θ), thenirank(RST(X,Y;θ))≥r(r - 1)^n-r+1/2.In Chapter 4,we introduce a new subsemigroup of T(X,Y; 0),that is order-preserving sandwich semigroup O(X,Y;θ).For finite order-preserving sandwich semigroup O(X,Y;θ), we consider the regularity,Green's relations,and some certain properties for it.The main results are given in following.Theorem 4.1.10(X,Y;θ)= Ox when X = Y andθ= 1x.Theorem 4.1.2 Let Oθ={αθ;α∈O(X,Y;θ)},then(1)Oθis a subsemigroup of Ox.(2)Oθ(?)O(X,Y;θ)ifθis injective.(3)Oθ= Ox and O(X,Y;θ)(?)Ox ifθis bijective.Theorem 4.2.1.2 For anyα,β∈O(X,Y;θ),α≠β,(α,β)∈(?)Xα= Yθα= Yθβ= Xβ.Theorem 4.2.1.3 For anyα,β∈O(X,Y;θ),α≠β,(α,β)∈(?)ker(α)= ker(β) and bothθ|xαandθ|xβare injective.Theorem 4.2.1.6 For anyα,β∈O(X,Y;θ),α≠β,then(α,β)∈D if and only if one of the following.(1)L_α=L_βand bothθ|xαandθ|Xβare not injective.(2)R_α= R_βand |Xα| = |Xβ|＞|Yθα| = |Yθβ|.(3)Bothθ|Xαandθ|Xβare injective and |Xα| = |Yθα| =|Yθβ|=|Xβ|.Theorem 4.2.2.2 For anyα∈O(X,Y;θ).α∈R(O(X,Y;θ))if and only if A∩Yθ≠φfor each A∈X/ker(α)andθ|xαis injective. Theorem 4.2.2.3 R(O(X,Y;θ))is a regular subsemigroup of O(X,Y;θ).Theorem 4.2.3.1 For anyα,β∈R(O(X,Y;θ)),α≠β,(α,β)∈L^R if and only if Xα= Xβ.Theorem 4.2.3.2 For anyα,β∈R(O(X,Y;θ)),α≠β,(α,β)∈R^R if and only if ker(α)=ker(β)Theorem 4.2.3.3 For anyα,β∈R(O(X,Y;θ)),α≠β,(α,β)∈D^R if and only if |Xα|= |Xβ|.In Chapter 5,we give some futher work.KeyWords E-order-preserving Transformations Semigroups,Sandwich Semigroup,Order-preserving Sandwich Semigroup2000 MR Subject Classification 20M10Chinese Library Classification O152.7...