关于Hamilton半群的研究
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摘要
本文给出Hamilton半群的基本性质,并且给出Hamilton半群的自同态半群与半直积,最后给出了Hamilton半群的自同构群和Hamilton半群的强右零带.具体内容如下:
第一章给出引言和预备知识.
第二章,首次给出Hamilton半群的子半群,同态像仍是Hamilton半群的证明,以及Hamilton半群对集合的作用.
主要结论如下:
定理2.1左(右)H-半群的子半群仍是左(右)H-半群.
定理2.3左(右)H-半群的同态像集合仍是左(右)H-半群.
定理2.6在左(右)H-半群S上,aρb (?) ak = b~l,其中,a,b∈S,k,l∈Z~+,则ρ是S上的最大幂等分离同余.
命题2.9在左H-半群S上,Green关系R是幂等纯同余;在左H-半群S上, Green关系L是幂等纯右同余.即R_e = E_S,L_e = E_S.命题2.17设左H-半群S,其中自同态半群EndS作用于S上,则x∈S的轨道x|-(?) {y∈S | y与x的指数相同}.
第三章给出了Hamilton半群的自同态半群也是Hamilton半群,并定义了降次Hamilton半群,讨论了Hamilton半群的自同态半群与降次Hamilton半群的半直积,直积.主要结论如下:
定理3.1任意在左(右)H-半群S的自同态集合EndS,关于乘法(fg)(x) = f(x)g(x),f,g∈End S,x∈S是一个左(右)H-半群.
定理3.6设S是任意一个左H-半群,S_1 = {a_i,i∈I},S_2 = {b_β,β∈A},S_3 = {e(p),p∈P},
In this dissertation, we characterize the Hamilton semigroup;be-sides ,we give a characterization of endomorphism semigroup and semi-direct product of Hamilton semigroup;finally,we give a definition andcharacterization of automorphism group and strong right zero band ofHamilton semigroup. The main results are given in follow.
In Chapter 1, we give the introduction and preliminaries.
In Chapter 2, we give the proof of subsemigroup and homo-morphic image which are still Hamilton semigroup;besides we discussthe action of Hamilton semigroup on a set .The main results are givenin follow.
Theorem 2.1 The subsemigroup of left (right) Hamilton semi-group is still left (right) Hamilton semigroup.
Theorem 2.3 The homomorphic image set of left (right) Hamil-ton semigroup is still left (right) Hamilton semigroup.
Theorem 2.6 On the left (right) Hamilton semigroup S,aρb (?)ak = bl,where a,b ∈ S,k,l ∈ Z+,then ρ is a maximum idempotent-separating congruence on S.
proposition 2.9 On the left Hamilton semigroup S,the R ofGreen relation is idempotent pure left congruence;On the left Hamil-ton semigroup S,the L of Green relation is idempotent pure right con-
gruence.That is Re = ES,Le = ES.In Chapter 3,we give the conclusion that the homomorphic imageof Hamilton semigroup is Hamilton semigroup;besides,we give a defi-nition descent order of Hamilton semigroup and discuss the semi-directproduct of the homomorphic semigroup of Hamilton semigroup and thedescent order of Hamilton semigroup.The main results are given in follow .Theorem 3.1 The the homomorphic set of Hamilton semigroupEndS on Hamilton semigroup S on the product(fg)(x) = f(x)g(x),f,g ∈ EndS,x ∈ S,is still a left (right) Hamilton semigroup.Theorem 3.6 S is a left Hamilton semigroup,S1 = {ai,i ∈ I},S2 = {bβ,β ∈ A},S3 = {e(p),p ∈ P},f is an endomorphism on the S,then(1)f |S1: S1 ?→ S, (2)f |S2: S2 ?→ S \ S1, (3)f |S3: S3 ?→ ESand be propitious to(4)f(aiaj) = f |S1 (ai)f |S1 (aj), ai,aj ∈ S,(5)f(bβai) = f |S2 (bβ)f |S1 (ai), ai,bβ ∈ S,(6)f(aibβ) = f |S1 (ai)f |S2 (bβ), ai,bβ ∈ S,(7)f(xe(p)) = f |Sk (x)f |S3 (e(p)), x,e(p) ∈ S,k ∈ {1,2,3}.is a map.On the other hand ,if f |S1,f |S2,f |S3 is a map as above,then f is ahomomorphism map.
Theorem 3.9 The semi-direct product EndS ×α S of endomor-phic semigoup EndS of the left Hamilton semigroup S and the descentorder of left Hamilton semigroup S is a left Hamilton semigroup, wherex ∈ S ,f ∈EndS,xf = f(x).Theorem 3.13 The direct product S × S of the descent orderof left Hamilton semigroup S of the left (right) Hamilton semigroup Sand itself is a left (right) Hamilton semigroup .In chapter 4,we mainly discuss automorphism of Hamilton semi-group is a group,what'more we give the decomposition of direct productof Hamilton semigroup;besides,we characterize the relation of the au-tomorphism of automorphism of Hamilton semigroup to twe Hamiltonsemigroup. The main results are given in follow.Theorem 4.1 Let S =i∈Iaiα∈Abαp∈Pep be a left Hamiltonsemigroup.AssumeS1 = {i∈Iai }, S2 = {α∈Abα }, S3 = {p∈Pep },AS1 ,AS2, AS3 are automorphism group of S1,S2,S3 individually,thenautomorphism group of AS automorphic AS1 ×AS2 ×AS3 .That is ASAS1 × AS2 × AS3.Theorem 4.7 If the automorphism group AS,AS of left (right)Hamilton semigroup S, S are automorphic,then S/ρ S /ρ.In chapter 5,we mainly discuss left Hamilton semigroup formata strong right zero band on the product and this strong zero band isa left Hamilton semigroup,beside the idempotents of format a strongright band.Theorem 5.1 Assume S is a a strong right band of Sα,α ∈
B,where B is a right band , Sα are all left Hamilton semigroup,then Sis left Hamilton semigroup.
第一章给出引言和预备知识.
第二章,首次给出Hamilton半群的子半群,同态像仍是Hamilton半群的证明,以及Hamilton半群对集合的作用.
主要结论如下:
定理2.1左(右)H-半群的子半群仍是左(右)H-半群.
定理2.3左(右)H-半群的同态像集合仍是左(右)H-半群.
定理2.6在左(右)H-半群S上,aρb (?) ak = b~l,其中,a,b∈S,k,l∈Z~+,则ρ是S上的最大幂等分离同余.
命题2.9在左H-半群S上,Green关系R是幂等纯同余;在左H-半群S上, Green关系L是幂等纯右同余.即R_e = E_S,L_e = E_S.命题2.17设左H-半群S,其中自同态半群EndS作用于S上,则x∈S的轨道x|-(?) {y∈S | y与x的指数相同}.
第三章给出了Hamilton半群的自同态半群也是Hamilton半群,并定义了降次Hamilton半群,讨论了Hamilton半群的自同态半群与降次Hamilton半群的半直积,直积.主要结论如下:
定理3.1任意在左(右)H-半群S的自同态集合EndS,关于乘法(fg)(x) = f(x)g(x),f,g∈End S,x∈S是一个左(右)H-半群.
定理3.6设S是任意一个左H-半群,S_1 = {a_i,i∈I},S_2 = {b_β,β∈A},S_3 = {e(p),p∈P},
In this dissertation, we characterize the Hamilton semigroup;be-sides ,we give a characterization of endomorphism semigroup and semi-direct product of Hamilton semigroup;finally,we give a definition andcharacterization of automorphism group and strong right zero band ofHamilton semigroup. The main results are given in follow.
In Chapter 1, we give the introduction and preliminaries.
In Chapter 2, we give the proof of subsemigroup and homo-morphic image which are still Hamilton semigroup;besides we discussthe action of Hamilton semigroup on a set .The main results are givenin follow.
Theorem 2.1 The subsemigroup of left (right) Hamilton semi-group is still left (right) Hamilton semigroup.
Theorem 2.3 The homomorphic image set of left (right) Hamil-ton semigroup is still left (right) Hamilton semigroup.
Theorem 2.6 On the left (right) Hamilton semigroup S,aρb (?)ak = bl,where a,b ∈ S,k,l ∈ Z+,then ρ is a maximum idempotent-separating congruence on S.
proposition 2.9 On the left Hamilton semigroup S,the R ofGreen relation is idempotent pure left congruence;On the left Hamil-ton semigroup S,the L of Green relation is idempotent pure right con-
gruence.That is Re = ES,Le = ES.In Chapter 3,we give the conclusion that the homomorphic imageof Hamilton semigroup is Hamilton semigroup;besides,we give a defi-nition descent order of Hamilton semigroup and discuss the semi-directproduct of the homomorphic semigroup of Hamilton semigroup and thedescent order of Hamilton semigroup.The main results are given in follow .Theorem 3.1 The the homomorphic set of Hamilton semigroupEndS on Hamilton semigroup S on the product(fg)(x) = f(x)g(x),f,g ∈ EndS,x ∈ S,is still a left (right) Hamilton semigroup.Theorem 3.6 S is a left Hamilton semigroup,S1 = {ai,i ∈ I},S2 = {bβ,β ∈ A},S3 = {e(p),p ∈ P},f is an endomorphism on the S,then(1)f |S1: S1 ?→ S, (2)f |S2: S2 ?→ S \ S1, (3)f |S3: S3 ?→ ESand be propitious to(4)f(aiaj) = f |S1 (ai)f |S1 (aj), ai,aj ∈ S,(5)f(bβai) = f |S2 (bβ)f |S1 (ai), ai,bβ ∈ S,(6)f(aibβ) = f |S1 (ai)f |S2 (bβ), ai,bβ ∈ S,(7)f(xe(p)) = f |Sk (x)f |S3 (e(p)), x,e(p) ∈ S,k ∈ {1,2,3}.is a map.On the other hand ,if f |S1,f |S2,f |S3 is a map as above,then f is ahomomorphism map.
Theorem 3.9 The semi-direct product EndS ×α S of endomor-phic semigoup EndS of the left Hamilton semigroup S and the descentorder of left Hamilton semigroup S is a left Hamilton semigroup, wherex ∈ S ,f ∈EndS,xf = f(x).Theorem 3.13 The direct product S × S of the descent orderof left Hamilton semigroup S of the left (right) Hamilton semigroup Sand itself is a left (right) Hamilton semigroup .In chapter 4,we mainly discuss automorphism of Hamilton semi-group is a group,what'more we give the decomposition of direct productof Hamilton semigroup;besides,we characterize the relation of the au-tomorphism of automorphism of Hamilton semigroup to twe Hamiltonsemigroup. The main results are given in follow.Theorem 4.1 Let S =i∈Iaiα∈Abαp∈Pep be a left Hamiltonsemigroup.AssumeS1 = {i∈Iai }, S2 = {α∈Abα }, S3 = {p∈Pep },AS1 ,AS2, AS3 are automorphism group of S1,S2,S3 individually,thenautomorphism group of AS automorphic AS1 ×AS2 ×AS3 .That is ASAS1 × AS2 × AS3.Theorem 4.7 If the automorphism group AS,AS of left (right)Hamilton semigroup S, S are automorphic,then S/ρ S /ρ.In chapter 5,we mainly discuss left Hamilton semigroup formata strong right zero band on the product and this strong zero band isa left Hamilton semigroup,beside the idempotents of format a strongright band.Theorem 5.1 Assume S is a a strong right band of Sα,α ∈
B,where B is a right band , Sα are all left Hamilton semigroup,then Sis left Hamilton semigroup.
引文
[1] 李师正, Hamilton 半群的结构, 纯粹数学与应用数学,9(1)(1993), 105-111.
[2] J.M.Howie, Fundamentals of Semigroup Theory, Oxford University Press Inc,Oxford, 1995.
[3] 刘绍学,每一子代数都是理想的代数, 数学学报, 14(4)(1964),532-537.
[4] A.H.Cliford and G.B Preston,The Algebraic Theory of Semigroups,Volume(II),(MathSurveys,Number 7,Amer.Math.Soc.1967).
[5] M.Hall,The theory of groups,New York,1959.
[6] G.M.S.Gomes,A Characterrization so the Group Congruences on a Semigroup,Semigroup Forum,46(1993),48-53.
[7] H.Mitsch and M.Petrich,Basic Properties of E-inversive Semigroups,Communicationsin Algebra,28(11), 5169-5182(2000).
[8] G.Chen,The Endomorphism Structure of Simple Faithful S-acts,SemigroupForum,59(1999),179-182.
[9] T.Evans,Properties of algebras almost equivalent to identities,J.LondonMath.Soc.37:1(1962)53-59.
[10] P.Y.Zhu, Y.Q.Guo and N.P.Shum, Structure and chacterizations of left C semi-groups, Science in China,Series A,6(1992):791-805.
[11] H.Hutchins,Division Semirings with 1 + 1=1,Semigroup Forum,22(1981),181- 188.
[12] D.R.LaTorre,Group Congruences on Regular Semigroups,Semigroup Forum,24(1982),327-340.
[13] T.W.Hungerford, 冯克勤译,聂灵沼校,代数学,湖南教育出版社,1985.
[14] 左孝凌,李为鉴,刘永才,离散数学,上海科学技术文献出版社,1982.
[15] J.E.Mills,Combinatorially Factorizable Inverse Monoids,Semigroup Forum,59(1999),220-232.
[16] J.Ahsan,Characterizations of Semirings By P-injective and ProjectiveSemimodules,Communications in Algebra,26(7), 2199-2209(1998).
[17] Petrich M.and Reilly N. ,Completely Regular Semigroups , 1999,John.Weley & Sonc.INC.
[18] M.Satyanarayana,on the Semigroup of Semirings,Semigroup ForumVol.23(1981)7-14.
[19] Chong-Yih Wu, On Right Congruences on Semilattices of Groups, SemigroupForum, 57(1998), 127-137.
[20] Petrich, M.,The structure of completely regular semigroups,Tran.Amer.Math.soc.1974(189):211-236.
[21] M.Petrich, Introduction to Semigroups, Merrill, Columbus, 1973.
[22] Petrich, M.,Lectures in Semigroups,Pitman:New York,1976.
[23] Fountain,J.B.,Abundant Semigroups,Proc.London Math.Soc.(3) 44(1982):103-129.
[24] H.Mitsch, Subdirect Products of E-inversive Semigroups, J.Austral. Math.Soc., 48(1990),66-78.
[25] D.A.Norton,Hamiltonian loop,proc.Amer.Math.Soc.3(1952),56-65
[26] P.G.Trotter,Covers for Regular Semigroups,SEA Bull.Math.,18 (3)(1994),157-161.
[2] J.M.Howie, Fundamentals of Semigroup Theory, Oxford University Press Inc,Oxford, 1995.
[3] 刘绍学,每一子代数都是理想的代数, 数学学报, 14(4)(1964),532-537.
[4] A.H.Cliford and G.B Preston,The Algebraic Theory of Semigroups,Volume(II),(MathSurveys,Number 7,Amer.Math.Soc.1967).
[5] M.Hall,The theory of groups,New York,1959.
[6] G.M.S.Gomes,A Characterrization so the Group Congruences on a Semigroup,Semigroup Forum,46(1993),48-53.
[7] H.Mitsch and M.Petrich,Basic Properties of E-inversive Semigroups,Communicationsin Algebra,28(11), 5169-5182(2000).
[8] G.Chen,The Endomorphism Structure of Simple Faithful S-acts,SemigroupForum,59(1999),179-182.
[9] T.Evans,Properties of algebras almost equivalent to identities,J.LondonMath.Soc.37:1(1962)53-59.
[10] P.Y.Zhu, Y.Q.Guo and N.P.Shum, Structure and chacterizations of left C semi-groups, Science in China,Series A,6(1992):791-805.
[11] H.Hutchins,Division Semirings with 1 + 1=1,Semigroup Forum,22(1981),181- 188.
[12] D.R.LaTorre,Group Congruences on Regular Semigroups,Semigroup Forum,24(1982),327-340.
[13] T.W.Hungerford, 冯克勤译,聂灵沼校,代数学,湖南教育出版社,1985.
[14] 左孝凌,李为鉴,刘永才,离散数学,上海科学技术文献出版社,1982.
[15] J.E.Mills,Combinatorially Factorizable Inverse Monoids,Semigroup Forum,59(1999),220-232.
[16] J.Ahsan,Characterizations of Semirings By P-injective and ProjectiveSemimodules,Communications in Algebra,26(7), 2199-2209(1998).
[17] Petrich M.and Reilly N. ,Completely Regular Semigroups , 1999,John.Weley & Sonc.INC.
[18] M.Satyanarayana,on the Semigroup of Semirings,Semigroup ForumVol.23(1981)7-14.
[19] Chong-Yih Wu, On Right Congruences on Semilattices of Groups, SemigroupForum, 57(1998), 127-137.
[20] Petrich, M.,The structure of completely regular semigroups,Tran.Amer.Math.soc.1974(189):211-236.
[21] M.Petrich, Introduction to Semigroups, Merrill, Columbus, 1973.
[22] Petrich, M.,Lectures in Semigroups,Pitman:New York,1976.
[23] Fountain,J.B.,Abundant Semigroups,Proc.London Math.Soc.(3) 44(1982):103-129.
[24] H.Mitsch, Subdirect Products of E-inversive Semigroups, J.Austral. Math.Soc., 48(1990),66-78.
[25] D.A.Norton,Hamiltonian loop,proc.Amer.Math.Soc.3(1952),56-65
[26] P.G.Trotter,Covers for Regular Semigroups,SEA Bull.Math.,18 (3)(1994),157-161.