半群CPOn(A)的格林关系
2015-05-26游泰杰
赵 颐,游泰杰
(贵州师范大学数学与计算机科学学院,贵阳 550001)
半群CPOn(A)的格林关系
赵 颐,游泰杰*
(贵州师范大学数学与计算机科学学院,贵阳 550001)
设POn是X n={1,2,…,n}上的保序部分变换半群,A是X n的非空子集,令CPOn(A)={α∈POn:(A∩dom(α))α⊆A,且∀x,y∈(A∩dom(α)),|xα-yα|≤|x-y|},则CPOn(A)是POn的子半群.利用变换半群的保序和压缩性,刻画了半群CPOn(A)的格林关系.
变换半群;保序部分变换;格林关系
在半群的众多分支中,变换半群是半群代数理论中一个重要研究方向,许多学者对部分变换半群Pn的各种子半群的格林关系进行了研究.Pei等[1-2]先后研究了保等价关系且保序变换半群、保等价关系变换半群的变种半群的格林关系;Sun等[3]研究了保等价关系且保方向变换半群的格林关系;Deng等[4-6]先后探讨了反向保等价关系变换半群、双向保等价关系变换半群及双向保等价关系且保序变换半群的格林关系;Zhao等[7]刻画了部分保序且压缩变换半群的格林关系;钟艳林等[8]给出了欧氏空间中升序变换半群的格林关系的一些刻画;Sangkhanan等[9]讨论了具有稳定值域的部分线性变换半群的格林关系.本文将研究半群CPOn(A)的格林关系,并给出若干等价刻画.
1 预备知识
设X n={1,2,…,n}且赋予自然序,A是X n的非空子集,Pn是X n上的部分变换半群.设α∈Pn,若对任意x,y∈dom(α),x≤y⇒xα≤yα,则称α是保序的.设POn是P n中所有保序部分变换之集(不含X n上的恒等变换),则POn是P n的子半群,并称POn为X n上的保序部分变换半群.令CPOn(A)={α∈POn:(A∩dom(α))α⊆A,且∀x,y∈(A∩dom(α)),|xα-yα|≤|x-y|},则易验证CPOn(A)是POn的子半群.
设S是半群,用S1表示在S上添加单位元.设a,b∈S,若a和b生成相同的主左理想,S1a=S1b,则称a与b是L等价的,并记为(a,b)∈L.类似可利用主右理想定义a与b是R等价的,并记为(a,b)∈R.本文未定义的术语及记法可参见文献[10].
2 半群CPOn(A)的格林关系
显然,若ker(α,A)=Ø,则ker(β,A)=Ø;若ker(α,A)={A k}(k∈{1,…,r}),则ker(β,A)={Bk}.现在,若|ker(α,A)|=t≥2,设ker(α,A)={Al1,A l2,…,Alt}(l1<l2<…<l t),则由(1)式可得ker(β,A)={Bl1,Bl2,…,Blt}.由δ,γ∈(CPOn(A))1,知(A∩dom(α))δ⊆A,(A∩dom(α))γ⊆A.注意到A∩Al1<A∩Al2<…<A∩Alt,A∩Bl1<A∩Bl2<…<A∩Blt,任取i,j∈{1,…,t}且i≤j,由Akδ⊆Bk,Bkγ⊆Ak,得
令d=max (A∩Al1)-max (A∩Bl1),则min (A∩A lt)-min (A∩Blt)=max (A∩A l1)-max(A∩Bl1)=d.若|ker(α,A)|=t≥3,则断言
若A li\A≠Ø,则任取x∈A li\A,设A∩A li中到点x距离最小者为.若Bli\A≠Ø,则任取y∈Ali\B,设A∩Bli中到点y距离最小者为.令
则由(6)式可得,α=δβ且β=γα.下面证明δ,γ∈CPOn(A).注意到
再由(6)式可得δ,γ∈POn,显然(A∩dom(δ))δ⊆A,(A∩dom(γ))γ⊆A.任取x,y∈A∩dom(δ)且x≤y.注意到A∩dom(δ)=(A∩A l1)∪(A∩A l2)∪…∪(A∩Alt),现分以下5种情形讨论.
情形1x,y∈A∩A li,i=1,t.显然xδ-yδ=0,因此|xδ-yδ|≤|x-y|.
情形2x∈A∩A l1,y∈A li,i∈[2,t-1].由(5),(7)式可得|xδ-yδ|=|max(A∩Bl1)-(yd)|=|y-(d+max(A∩Bl1))|=|y-max(A∩A l1)|≤|x-y|.
情形3x∈A∩Al1,y∈A∩Alt.由(5),(7)式可得|xδ-yδ|=|max(A∩Bl1)-min(A∩Blt)|=max (A∩Al1)-min (A∩Alt) ≤|x-y|.
情形4x∈A∩Ali,y∈A∩Alj,i,j∈[2,t-1].显然,|xδ-yδ|=|(x-d)-(y-d)|=|x-y|,因此|xδ-yδ|≤|x-y|.
情形5x∈A∩A l1,i∈[2,t-1],y∈A∩Alt.由(5),(7)式可得|xδ-yδ|=|(x-d)-min(A∩Blt)|= min (A∩Blt)-(x-d) = (min (A∩Blt)+d)-x= min (A∩A lt)-x≤|x-y|.
综上,证得δ∈CPOn(A).同理可证,γ∈CPOn(A);因此,(α,β)∈L.
定理3 设α,β∈CPOn(A),则(α,β)∈D当且仅当
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Green’s relations on the semigroupCPOn(A)
ZHAO Yi,YOU Taijie*
(Sch of Math &Comput Sci,Guizhou Norm Univ,Guiyang 550001,China)
LetPOnbe the partial order-preserving transformation semigroup onX n={1,…,n}.For each nonempty subsetAofXn,letCPOn(A)= {α∈POn:(A∩dom(α))α⊆A,∀x,y∈(A∩dom(α)),|xα-yα|≤|x-y|}.ThenCPOn(A)is a subsemigroup ofPOn.In this paper,using order-preserving and compression properties of the transformation semigroup,Green’s relations on the semigroupCPOn(A)are characterized.
transformation semigroup;partial order-preserving transformation;Green’s relations
O 152.7
A
1007-824X(2015)04-0005-04
2015-07-02.* 联系人,E-mail:youtaijie1959@163.com.
国家自然科学基金资助项目(11461014).
赵颐,游泰杰.半群CPOn(A)的格林关系 [J].扬州大学学报(自然科学版),2015,18(4):5-8,12.
(责任编辑 青 禾)