Three Kinds Of Semigroups That Preserve An Equivalence

Semigroup theory is a young subject,It has been deeply researched. Semigroup theory has applied in many fields, such as computer science , automata theory, theory of coding, cryptography, and so on. it is appearing strong vigor and it would has more and more applications.In semigroup theory, transformatiom semigroup was always a active task in last decades. since each semigroup can be inseted in some transformation semigroup. Theoretically, for abstract semigroup theory it is enough to study transformation semigroup .Finite full transformation semigroup has attacted many scholars thanks for it is computable and it has many well combinatorial properties. Green's relations, first studied by J.A.Green in 1951, have played a fundamental role in the development of semigroup, especially, in the development of finite full transformation semigroup. Many schoraies have researched deeply for full transformation semigroup and its some subsemigroups. For example, Hoiwe, You [11-14], Pei[6-9], Yang [15-16], and so on.All continues selfmaps of a given topological spaces X make up of a semigroup under composition. Green's relation for it's regular elements were characterized by Magill and Subiah(1974). Recently, Pei [8,2005] observed a class of topological spaces: if E is an equivalence on a set X. let X be endowed with the topology for which the family of all E-classes is a basis. let T_e(X) = {α∈T_X :(?)(x,y)∈E (?)(xα,yα)∈E} and has researched the regularities,the Green's relations and some other properties of T_E(X) when |X| is finite.Denote By T_X the full transformation semigroup on X, E is an equivalence on a set X. In Chapter 2, a new kind of subsemigroups of T_X, Semigroup of transformations that preserve double direction equivalence T_e^*(X) is considered.T_E^*(X) = {α∈T_X :(?)(x, y)∈E(?)(xα, yα)∈E}Green's relations of T_E^*(X) are discussed in Section 2.2, the regularity of T_E^*(X) is considered in Section 2.3In Chapter 3, a new kind of subsemigroups of T_X, Semigroup of transformations that preserve double direction equivalence T_?(X) is considered.T_?(X) = {α∈T_X: (?)(xα, yα)∈E(?) (x, y)∈E}Green's relations of T_?(X) are discussed in Section 3.2, the regularity of T_?(X) is considered in Section 3.3 In Chapter 4, a kind of matrix semigroup that preserve an equivalence is considered. letα=(?),β=(?)∈R^2×1denoteα～β(?)m+n=s+tobviously the～is an equivelance between vectors on plane. LetS = {A∈R^2×2|(?)α,β∈R^2×1,α～βthen Aα～Aβ}Clearly S is a semigroup under composition. Green's relations of S are discussed in Section 4.2, the idempotentence and regularity of S are considered in Section 4.3.The main results are given as following.In Chapter 1. we give some basic conseption about semigroup theory.Theorem 2.1.3 (1)E = X×X if and only if T_E^*(X) = T_X; (2)E = I( the identity relation on X) if and only if T_E^*(X) = {α∈T_X|(?)x,y∈X,x≠y(?)xα≠yα}.Theorem 2.2.1 For anyα,β∈T_E^*(X), the following are equivalent(1) (α,β)∈L;(2) Xα= Xβ;(3) There is an inadmissible bijection (?):Ï€(α)→π(β) such thatα_* = (?)β_*.Theorem 2.2.2 Forα,β∈T_E^*(X), the following are equivalent(1) (α,β)∈R;(2)Ï€(α) =Ï€(β);|Z(α)| = |Z(β)|(3) There existsδ∈T_E^*(X),δ|_Xα:Xα→Xβis a bijection andβ=αδ. There existsσ∈T_E^*(X),σ|_Xβ:Xβ→Xαis a bijection andα=βσ.Theorem 2.2.3 Forα,β∈T_E^*(X), the following are equivalent(1) (α,β)∈H;(2) Xα= Xβ,Ï€(α) =Ï€(β);(3) There is an E^*-admissible bijection (?) :Ï€(α)→π(β) such thatα_* = (?)β_*.There areδ,σ∈T_E^*(X) such thatδ|_Xα:Xα→Xβ,σ|_Xβ:Xβ→Xαare bijections andβ=αδ,α=βσ.Theorem 2.2.4 Forα,β∈T_E^*(X), the following are equivalent (1) (α,β)∈D; (2) |Z(α)| = |Z(β)| There existsδ∈T_E^*(X),δ|_Xα : Xα→Xβis a bijection.Theorem 2.2.5 Forα,β∈T_E^*(X), the following are equivalent(1) (α,β)∈J;(2) |Xα| = |Xβ|, There existρ,τ∈TT_E^*(X) for any A∈X/E, Aα(?)BβρAβ(?)Cατfor some B,C∈X/ETheorem 2.2.6 Letα,β∈T_E^*(X). |X/E| = n (n is a integer). Then the following statements are equivalent:(1)(α,β)∈D.(2)(α,β)∈J(3) |Xα| = |Xβ|. there existρ,τ∈T_E^*(X) , for any A∈X/E, Aα(?)Bβρ, Aβ(?)Cατfor some B,C∈X/E.Theorem 2.3.1 Letα∈T_E^*(X), thenαis regular if and only if A∩Xα≠(?) for anyA∈X/E.Theorem 2.3.2 T_E^*(X) is regular if and only of |X/E| is finite.Theorem 2.3.3 Letα,β∈T_E^*(X). (αβ)² =αβthenα,βare regular.Theorem 3.1.4(1)T_E^*(X)=T_E(X)∩T_?(X).(2) If |X/E| =n(n is a integer), then T_E^*(X) = T_?(X).(3) E = X×X if and only if T_?(X) = T_X.(4)E = I if and only if T_?(X) = {α∈T_X : (?)x,y∈X,x≠y(?)xα≠yα}.Theorem 3.2.1 Letα,β∈T_?(X). Then the following statements are equivalent:(1) (α,β)∈L.(2) Xα= Xβand for each A∈X/E there exist B, C∈X/E such that Aα= Bβand Aβ= Cα(3) There is an E^* -admissible bijection (?) :Ï€(α)→π(β) such thatα_* = (?)β_*.Theorem 3.2.2 Letα,β∈T_?(X). Then the following statements are equivalent:(1) (α,β)∈R.(2)Ï€(α) =Ï€(β). |Z(α)| = |Z(β)|. and for any x,y∈X, (xα,yα)∈E if and only if (xβ,yβ)∈E(3) There existsδ∈T_E^*(X) such thatδ|_Xα : Xα→Xβis bijection andβ=αδ. There existsσ∈T_E^*(X) such thatδ|_Xβ:Xβ→Xαis bijection andα=βσ.Theorem 3.2.3 Letα,β∈T_E^*(X). Then the following statements are equivalent: (1)(α,β)∈H．(2)Xα=Xβand for each A∈X/E there exist B,C∈C/E such that Aα=Bβand Aβ=Cα．π(α)=Ï€(β) and for any x,y∈X,(xα,yα)∈E if and only if(xβ,yβ)∈E (3)There is an E^*-admissible bijection (?):Ï€(α)→π(β)such thatα_*=(?)β．There areδ,σ∈T_E^*(X) such thatδ|_Xα:Xα→Xβ,σ|_Xβ:Xβ→Xα．are bijectionsandβ=αδ,α=βσ.Theorem 3．2．4 Letα,β∈T_?(X)．Then the following statements are equivalent:(1)(α,β)∈D．(2)|Z(α)|=|Z(β)|．there existsδ∈T_E^*(X) such thatδ|_Xα:Xα→Xβis bijective．for each A∈X／E there exist B,C∈X／E such that Aαδ=Bβ,Aβ=CαδLemma 3．2．5 Letα,β∈T_?(X),suppose |Z(β)|≤|Z(α)| and there exists an E^*-mappingδ:Xβ→Xαsuch that for each A∈X/E,Aα=Bβδfor single B∈X/E．thenα=θβηfor someθ,η∈T_?(X)．Corollary 3．2．7 Letα,β∈T_?(X)and there existδ,σ∈T_E^*(X)such that for eachA∈X／E,Aα=Bβδ,Aβ=Cασfor single B,C∈X／E．then(α,β)∈J．Theorem 3．2．8 Letα,β∈T_?(X)．|X／E|=n(n is a integer)．Then the followingstatements are equivalent: (1)(α,β)∈D．(2)(α,β)∈J (3)|Xα|=|Xβ|.there existρ,τ∈T_E^*(X),for any A∈X／E,Aα(?)Bβρ,Aβ(?)Cατfor some B,C∈X/E．Theorem 3．3．1 Letα∈T_?(X)．thenαis regular if and only if Aα∈Xα/E_αandA∩Xα≠(?) for each A∈X／E．Theorem 3．3．2 T_?(X)is regular if and only if |X／E| is,finiteTheorem 3．3．3 Letα,β∈T_?(X)．(αβ)²=αβthenα,βare regular．Theorem 4．1．1 A=(?)∈S if and only if a₁₁+a₂₁=a₁₂+a₂₂．Corollary 4．1．3 {A∈S:det(A)≠0)is a subgroup of S．Corollary 4．1．4 {A∈S:det(A)=0) is a subsemigroup S．Theorem 4．2．1 For A,B∈uinv(S)ï¼›(A,B)∈L if and only if one of fouowingstatements holds. (1)A_r=B_r=1,A_c=B_c=-1．(2)A_r=B_r=1,A_c≠-1,B_c≠-1．(3)A_r=B_r≠1,A_c=B_c=-1．Theorem 4．2．2 For A,B∈uinv(S)ï¼›(A,B)∈R if and only if one of fouowingstatements holds．．(1)A_c=B_c=-1,A_r=B_r=1．(2)A_c=B_c=-1,A_r≠1,B_r≠1．(3)A_c=B_c≠-1,A_r=B_r=1．Theorem 4．2．3 For A,B∈uinv(S)ï¼›(A,B)∈H if and only if A_r=B_r,A_c=B_c．Theorem 4．2．4 For A,B∈uinv(S)ï¼›(A,B)∈D if and only if one of followingstatements holds．(1)A_r=B_r=1,A_c=-1,B_c=-1．(2)A_r=B_r=1,A_c≠-1,B_c≠-1．(3)A_c=B_c=-1,A_r≠1,B_r≠1．Theorem 4．2．5 For A,B∈uinv(S)ï¼›(A,B)∈J if and only if one of followingstatements holds．(1)A_r=B_r=1,A_c=-1,B_c=-1．(2)A_r=B_r=1,A_c≠-1,B_c≠-1．(3)A_c=B_c=-1,A_r≠1,B_r≠1．Theorem 4．3．1 Let A∈S^*,then A²=A,if and only if one of the following statements holds．(1):A=E．(2):A=(?)(a≠0)．(3):A=(?)(a≠0).Theorem 4．3．2 Let A∈S^*,then A is regular if and only if oile of the followingstatements holds．(1)det(A)≠0．(2)A∈uinv(S)and A_r=1,A_c≠-1 (3)A∈uinv(S)and A_r≠1,A_c=-1In Chapter 5,we give some further work．...