Clifford半群的推广
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摘要
在半群代数理论中，正则半群的研究一直占据主导地位。Clifford半群作为一类重要的正则半群，早在1941年，Clifford就开始了对此类半群的研究，并且给出了它的一个优美的结构定理。1991年，朱聘瑜，郭聿琦和岑嘉评在正则半群范围内，对Clifford半群进行了推广，定义了所谓左C-半群，不仅对左C-半群的特征进行了刻画，而且给出了左C-半群的ξ-积结构。之后，1995年，郭聿琦，任学明和岑嘉评又给出了左C-半群的一个新的结构，所谓Δ-积结构。另一方面，在1994年，任学明，郭聿琦和岑嘉评还在拟正则半群内，对Clifford半群进行了推广，定义了Clifford拟正则半群，并建立了它的θ-积结构。
在上述研究的基础上，本文首先研究了左C-半群的左交错积结构，证明了左正则带和Clifford半群的左交错积恰为一左C-半群；反过来，任意左C-半群都可以表示为一个左正则带和Clifford半群的左交错积，并且刻画了它的两种特殊情形。其次，研究了Clifford拟正则半群上的同余理论。首先引入了中心同余对的概念，利用Clifford拟正则半群上的中心同余对，证明了Clifford拟正则半群上的任何同余都可由一个中心同余对惟一表示。
The study of regular semigroups is always an important part in the theory of semigroups. The class of Clifford semigroups is an important subclass of the class of regular semigroups. In 1941, Clifford first studied these semigroups and gave a fine structure theorem for Clifford semigroups. Later on, P.Y.Zhu, Y.Q.Guo and K.P.Shum extended Clifford semigroups in the class of regular semigroups and gave the definition of left C-semigroups in 1991. In the same time, the basic characteristics of left C-semigroups were researched and the ζ-product of left C-semigroups was described. In 1995, Y.Q.Guo, X.M.Ren and K.P.Shum also built a new structure of left C-semigroups, that is, △-product. On the other hand, X.M.Ren, Y.Q.Guo and K.P.Shum extended Clifford semigroups to the class of quasiregular semigroups in 1994. They defined Clifford quasiregular semigroups and gave a θ-product structure of this kind of semigroups.
In this thesis, the left cross product of left C-semigroups is investigated. It is proved that the left cross product of a left regular band and a Clifford semigroup is the left C-semigroup; Conversely, any left C-semigroup can be constructed by using the left cross product of a left regular band and a Clifford semigroup. And their two special situations are considered. Secondly, we research the congruence on Clifford quasiregular semigroups by using central congruence pairs on Clifford quasiregular semigroups. It is proved that any congruences on Clifford quasiregular semigroups can be unique expressed by central congruence pairs.
在上述研究的基础上，本文首先研究了左C-半群的左交错积结构，证明了左正则带和Clifford半群的左交错积恰为一左C-半群；反过来，任意左C-半群都可以表示为一个左正则带和Clifford半群的左交错积，并且刻画了它的两种特殊情形。其次，研究了Clifford拟正则半群上的同余理论。首先引入了中心同余对的概念，利用Clifford拟正则半群上的中心同余对，证明了Clifford拟正则半群上的任何同余都可由一个中心同余对惟一表示。
The study of regular semigroups is always an important part in the theory of semigroups. The class of Clifford semigroups is an important subclass of the class of regular semigroups. In 1941, Clifford first studied these semigroups and gave a fine structure theorem for Clifford semigroups. Later on, P.Y.Zhu, Y.Q.Guo and K.P.Shum extended Clifford semigroups in the class of regular semigroups and gave the definition of left C-semigroups in 1991. In the same time, the basic characteristics of left C-semigroups were researched and the ζ-product of left C-semigroups was described. In 1995, Y.Q.Guo, X.M.Ren and K.P.Shum also built a new structure of left C-semigroups, that is, △-product. On the other hand, X.M.Ren, Y.Q.Guo and K.P.Shum extended Clifford semigroups to the class of quasiregular semigroups in 1994. They defined Clifford quasiregular semigroups and gave a θ-product structure of this kind of semigroups.
In this thesis, the left cross product of left C-semigroups is investigated. It is proved that the left cross product of a left regular band and a Clifford semigroup is the left C-semigroup; Conversely, any left C-semigroup can be constructed by using the left cross product of a left regular band and a Clifford semigroup. And their two special situations are considered. Secondly, we research the congruence on Clifford quasiregular semigroups by using central congruence pairs on Clifford quasiregular semigroups. It is proved that any congruences on Clifford quasiregular semigroups can be unique expressed by central congruence pairs.
引文
[1] 朱聘瑜，郭聿琦，岑嘉评，左Clifford 半群的特征与结构．中国科学(A辑)，1991，6：582-590．
[2] Petrich. M, Inverse semigroups. John Wiley and Sons, New York, 1984.
[3] 郭聿琦，任学明，岑嘉评，左C-半群的又一结构．数学进展，1995，第一期，第24卷：39-43．
[4] S. Bogdanovic, Semigroups with a system of subsemigroups. Yugoslavia, Novi Sad, 1986.
[5] J. M. Howie, An introduction to semigroups theory. Academic press, London, 1976.
[6] A. H. Cifford, The constructure of orthodox unions of groups. Semigroup Forum, 1972, 3: 283-337.
[7] P. Jones, Congruence semimodular varieties of semigroups. Lecture Notes in Math, Springer, Berlin-New York, 1988, 1320: 162-171.
[8] J. M. Howie, Fundamentals of semigroup theory. Oxford University Press, New York, 1995.
[9] A. E1 Qallali, L*-unipotent semigroup. Pure and Applied Algebra, 1989, 62: 19-33.
[10] X. M. Ren, K. P. Shum, Y. Q. Guo, On super R*-unipotent semigroups. Southeast Asian Bulletin of Mathematics, 1998, 22: 199-208.
[11] M. Petrich, The structure of completely regular semigroups. Tran. Amer. Math, Soc, 1976, 189: 211-236.
[12] M. Petrich, Congruences on completely regular semigroups. Can. J. Math, 1989, XLI. 3: 439-461.
[13] M. Petrich, Congruences on extension of semigroups. Duke Math, 1967, 34: 215-224.
[14] J. B. Fountain, Right PP monoids with center idempotent. Semigroup Forum, 1977, 13: 229-237.
[15] X. J. Guo, Y. Q. Guo, K. P. Shum, Semi-spined structure of left C-a semigroups. Semigoups, Springer-Verlag, Singapore, 1998, 157-166.
[16] Y. Q. Guo, K. P. Shum, P. Y. Zhu, The structure of left C-rpp semigroups. Semi-group Forum, 1995, 50: 9-23.
[17] Tang Xiangdong, On a theorem of C-wrpp semigroup. Communications in Al-gebra, 1997, 25(5): 1499-1504.
[18] Lan Du, K. P. Shum, On left C-wrpp semigroup. Semigroup Forum, 2003, 67: 373-387.
[19] Tang Xiangdong, Semilattices of L**-simple semigroup. Semigroup Forum, 1998, 57: 37-41.
[20] K. P. Shum, X. M. Ren, Abundant semigroups and their special subclasses. Pro-ceedings of the International Conference on Algebra and its Applications, 2002, 66-86.
[21] K. P. Shum, Y. Q. Guo, X. M. Ren, Admissible congruence pairs on quasiregualer semigroups. Algebras groups and geometries, 1999, 16: 127-144.
[22] X. M. Ren, K. P. Shum, Structure theorems for right pp-semigroups with left cen-tral idempotents. General Algebra and Applications, 2000, 20: 63-75.
[23] 郭聿琦，岑嘉评，朱聘瑜，左C-rpp 半群．科学通报，1992，37(4)：492-494．
[24] 朱平，郭聿琦，任学明，拟完全纯整半群到拟群的半格(矩阵)-矩阵(半格)分解．数学年刊，1989，10A：6，742-752．
[25] 杜兰，关于毕竟C-wrpp半群．宁夏大学学报(自然科学版)，2001，22(1)：9-11．
[26] Tang Xingdong, A note on C-wrpp semigroup. Northeast. Math. 2001, 17(1): 71-74.
[27] Y. Q. Guo, The structure of weakly left C-semigroups. Chinese Science Bulletin, 1995, 40: 1744-1747.
[28] 任学明，岑嘉评，超富足半群的结构．中国科学(A辑)，2003，33(6)：551-561．
[29] A. EI-Qallali, J. B. Fountain, Idempotent-connected abundant semigroup. Proc. Roy. Soc. Edinburgh Sect. A, 1981, 91: 79-90.
[30] P. G. Trotter, Congruence on regular and completely semigroups. Austral Math Soc. SerA, 1982, 32: 388-398.
[31] 瑜秉钧，严格π-正则半群的结构．中国科学(A辑)，1990，11：1159-1161．
[32] 郭聿琦，任学明，E-理想拟正则半群．中国科学(A辑)，1989，5：479-486．
[33] J. B. Fountain, Abundant semigroups. Proc. London. Math Soc . , 1982, 44(3): 103-129.
[34] J. B. Fountain, Adequate semigroups. Proc. Edinburgh Math Soc. , 1979, 22: 113-125.
[35] Gang Li, Y. Q. Guo, K. P. Shum, Quasi-C-Ehresmann semigroups and their sub-class. Semigroup Forum, 2005, 1-22.
[36] 任学明，郭聿琦，岑嘉评，Clifford拟正则半群．数学年刊，1994，15A(3)：319-325．
[37] K. P. Shum, X. M. Ren, Abundant semigroups and their special subclasses. Pro-ceedings of the International Conference on Algebra and its Applications, 2002.
[38] R. J. Warne, On the structure of TC semigroups. In" Semigroups, Algebraic theory and applications to formal languages and codes " , Eds. C. Bonzini, A. Cherubini, C. Tibiletti (World Sicentific), 1993, 300-310.
[2] Petrich. M, Inverse semigroups. John Wiley and Sons, New York, 1984.
[3] 郭聿琦，任学明，岑嘉评，左C-半群的又一结构．数学进展，1995，第一期，第24卷：39-43．
[4] S. Bogdanovic, Semigroups with a system of subsemigroups. Yugoslavia, Novi Sad, 1986.
[5] J. M. Howie, An introduction to semigroups theory. Academic press, London, 1976.
[6] A. H. Cifford, The constructure of orthodox unions of groups. Semigroup Forum, 1972, 3: 283-337.
[7] P. Jones, Congruence semimodular varieties of semigroups. Lecture Notes in Math, Springer, Berlin-New York, 1988, 1320: 162-171.
[8] J. M. Howie, Fundamentals of semigroup theory. Oxford University Press, New York, 1995.
[9] A. E1 Qallali, L*-unipotent semigroup. Pure and Applied Algebra, 1989, 62: 19-33.
[10] X. M. Ren, K. P. Shum, Y. Q. Guo, On super R*-unipotent semigroups. Southeast Asian Bulletin of Mathematics, 1998, 22: 199-208.
[11] M. Petrich, The structure of completely regular semigroups. Tran. Amer. Math, Soc, 1976, 189: 211-236.
[12] M. Petrich, Congruences on completely regular semigroups. Can. J. Math, 1989, XLI. 3: 439-461.
[13] M. Petrich, Congruences on extension of semigroups. Duke Math, 1967, 34: 215-224.
[14] J. B. Fountain, Right PP monoids with center idempotent. Semigroup Forum, 1977, 13: 229-237.
[15] X. J. Guo, Y. Q. Guo, K. P. Shum, Semi-spined structure of left C-a semigroups. Semigoups, Springer-Verlag, Singapore, 1998, 157-166.
[16] Y. Q. Guo, K. P. Shum, P. Y. Zhu, The structure of left C-rpp semigroups. Semi-group Forum, 1995, 50: 9-23.
[17] Tang Xiangdong, On a theorem of C-wrpp semigroup. Communications in Al-gebra, 1997, 25(5): 1499-1504.
[18] Lan Du, K. P. Shum, On left C-wrpp semigroup. Semigroup Forum, 2003, 67: 373-387.
[19] Tang Xiangdong, Semilattices of L**-simple semigroup. Semigroup Forum, 1998, 57: 37-41.
[20] K. P. Shum, X. M. Ren, Abundant semigroups and their special subclasses. Pro-ceedings of the International Conference on Algebra and its Applications, 2002, 66-86.
[21] K. P. Shum, Y. Q. Guo, X. M. Ren, Admissible congruence pairs on quasiregualer semigroups. Algebras groups and geometries, 1999, 16: 127-144.
[22] X. M. Ren, K. P. Shum, Structure theorems for right pp-semigroups with left cen-tral idempotents. General Algebra and Applications, 2000, 20: 63-75.
[23] 郭聿琦，岑嘉评，朱聘瑜，左C-rpp 半群．科学通报，1992，37(4)：492-494．
[24] 朱平，郭聿琦，任学明，拟完全纯整半群到拟群的半格(矩阵)-矩阵(半格)分解．数学年刊，1989，10A：6，742-752．
[25] 杜兰，关于毕竟C-wrpp半群．宁夏大学学报(自然科学版)，2001，22(1)：9-11．
[26] Tang Xingdong, A note on C-wrpp semigroup. Northeast. Math. 2001, 17(1): 71-74.
[27] Y. Q. Guo, The structure of weakly left C-semigroups. Chinese Science Bulletin, 1995, 40: 1744-1747.
[28] 任学明，岑嘉评，超富足半群的结构．中国科学(A辑)，2003，33(6)：551-561．
[29] A. EI-Qallali, J. B. Fountain, Idempotent-connected abundant semigroup. Proc. Roy. Soc. Edinburgh Sect. A, 1981, 91: 79-90.
[30] P. G. Trotter, Congruence on regular and completely semigroups. Austral Math Soc. SerA, 1982, 32: 388-398.
[31] 瑜秉钧，严格π-正则半群的结构．中国科学(A辑)，1990，11：1159-1161．
[32] 郭聿琦，任学明，E-理想拟正则半群．中国科学(A辑)，1989，5：479-486．
[33] J. B. Fountain, Abundant semigroups. Proc. London. Math Soc . , 1982, 44(3): 103-129.
[34] J. B. Fountain, Adequate semigroups. Proc. Edinburgh Math Soc. , 1979, 22: 113-125.
[35] Gang Li, Y. Q. Guo, K. P. Shum, Quasi-C-Ehresmann semigroups and their sub-class. Semigroup Forum, 2005, 1-22.
[36] 任学明，郭聿琦，岑嘉评，Clifford拟正则半群．数学年刊，1994，15A(3)：319-325．
[37] K. P. Shum, X. M. Ren, Abundant semigroups and their special subclasses. Pro-ceedings of the International Conference on Algebra and its Applications, 2002.
[38] R. J. Warne, On the structure of TC semigroups. In" Semigroups, Algebraic theory and applications to formal languages and codes " , Eds. C. Bonzini, A. Cherubini, C. Tibiletti (World Sicentific), 1993, 300-310.