半群M(n,k)的正则性和格林关系
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摘要
设X_n={1,2,…,n}为有限链,T_n是X_n上的全变换半群。给定k∈X_n,记W(n,k),R(n,k)分别为T_n的如下子集{f∈T_n:(x,y∈X_n),|x-k|≤|y-k|?|f(x)-k|≤|f(y)-k|},{f∈T_n:(x,y∈X_n),|x-k|≤|y-k|?|f(x)-k|≥||f(y)-k||}W(n,k)与R(n,k)的并集记作M(n,k)。显然,M(n,k)是T_n的子半群。讨论了半群M(n,k)的正则性并刻画了它的格林关系。
Let X_n= { 1,2,…,n} be a finite chain and let T_n be the full transformation semigroup on X_n. Suppose that k is a fixed element of X_n,denote by W(n,k) and R(n,k) the subsets of T_ndefined as{f ∈ T_n:(x,y ∈ X_n),|x-k| ≤ |y-k|?|f(x)-k| ≤ |f(y)-k|},{f∈T_n:(x,y ∈ X_n),|x-k| ≤ |y-k|?|f(x)-k| ≥ |f(y)-k|}respectively,and denote by M(n,k) the union of W(n,k) and R(n,k). Obviously,M(n,k) is a subsemigroup of T_n. In this paper,we discuss the regularity of the semigroup M(n,k) and characterize its Green's relations.
Let X_n= { 1,2,…,n} be a finite chain and let T_n be the full transformation semigroup on X_n. Suppose that k is a fixed element of X_n,denote by W(n,k) and R(n,k) the subsets of T_ndefined as{f ∈ T_n:(x,y ∈ X_n),|x-k| ≤ |y-k|?|f(x)-k| ≤ |f(y)-k|},{f∈T_n:(x,y ∈ X_n),|x-k| ≤ |y-k|?|f(x)-k| ≥ |f(y)-k|}respectively,and denote by M(n,k) the union of W(n,k) and R(n,k). Obviously,M(n,k) is a subsemigroup of T_n. In this paper,we discuss the regularity of the semigroup M(n,k) and characterize its Green's relations.
引文
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[2]GARBA G U.On the idempotent ranks of certain semigroups of order-preserving transformations[J].Portugaliae Mathematica,1994,51(2):185-204.
[3]SOMMANEE W,SANWONG J.Order-preserving transformations with restricted range:Regularity,Green’s relations,and ideals[J].Algebra Universalis,2015,74(3-4):277-291.
[4]ZHANG J,LUO Y F.On certain order-preserving transformations semigroups[J].Communications in Algebra,2017,7:2980-2995.
[5]FERNANDES V H,KOPPITZ J,MUSUNTHIA T.The rank of the semigroup of all order-preserving transformations on a finite fence[J].Bulletin of the Malaysian Mathematical Science Society,2018,201(11):1-21.
[6]PEI H S.Regularity and Green's relations for semigroups of transformations that preserve an equivalence[J].Communications in Algebra,2005,33(1):109-118.
[7]SUN L.Regularity and Green's relations for semigroups of transformations preservings orientation and an equivalence[J].Semigroup Forum Vol,2007,74(2007):473-486.
[8]SUN L,Huisheng Pei.Green's relations for the variants of semigroups transformations preservings an equivalence relation[J].Communications in Algebra,2007,35:1971-1986.
[9]SUN L,Huisheng Pei.Naturally ordered transformations semigroups preserving an equivalence[J].Bull of the Austral Mathematical Society,2008,117-128.
[10]SUN L,XIN X J.The naturally partial order on the semigroup of all transformations of a set that reflect an equivalence relation[J].Bull of the Austral Mathematical Society,2013,88:359-368.
[11]裴惠生.保持一个等价关系的部分一一变换半群的Green关系[J].信阳师范学院学报(自然科学版),2017,30(1):1-4.
[12]龙伟锋,龙伟芳,高荣海.一类保等价关系部分变换半群的Green关系和正则性[J].贵州师范大学学报(自然科学版),2008,26(4):73-77.
[13]秦美青,许新斋.保持等价部分变换半群的变种半群上的正则元[J].纯粹数学与应用数学,2010,26(5):822-827.
[14]裴惠生,邓伟娜.保持双向等价关系的变换半群的若干结果[J].西南大学学报(自然科学版),2012,34(8):6-10.
[15]HOWIE J M.Fundamentals of semigroup theory[M].London:Oxford Press,1995.