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ArticleOriginal scientific text

Title

Green's relations and their generalizations on semigroups

Authors 1, 2, 3

Affiliations

  1. Department of Mathematics, The University of Hong Kong, Hong Kong, China (SAR)
  2. Department of Mathematics, Ningxia University, Yinchuan, Ningxia, China
  3. Institute of Mathematics, Southwest University, Beibei, Chongqing, China

Abstract

Green's relations and their generalizations on semigroups are useful in studying regular semigroups and their generalizations. In this paper, we first give a brief survey of this topic. We then give some examples to illustrate some special properties of generalized Green's relations which are related to completely regular semigroups and abundant semigroups.

Keywords

ENG
Green's relations, generalized Green's relations, regular semigroups, abundant semigroups rpp-semigroups

Bibliography

  1. A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, Vol. I. Mathematical Surveys, No. 7 American Mathematical Society, Providence, R.I. 1961 xv+224 pp.
  2. A.H. Clifford and G.B. Preston, The algebraic theory of semigroups, Vol. II., Mathematical Surveys, No. 7, American Mathematical Society, Providence, R.I. 1967 xv+350 pp.
  3. A. El-Qallali and J.B. Fountain, Idempotent-connected abundant semigroups, Proc. Roy. Soc. Edinburgh Sect. A 91 (1-2) (1981/82), 79-90.
  4. A. El-Qallali and J.B. Fountain, Quasi-adequate semigroups, Proc. Roy. Soc. Edinburgh Sect. A 91 (1-2) (1981/82), 91-99.
  5. J.B. Fountain, A class of right PP monoids, Quart. J. Math. Oxford Ser. 2, 28 (111) (1977) 285-300.
  6. J.B. Fountain, Abundant semigroups, Proc. London Math. Soc. 44 (3) (1982), 103-129.
  7. J.B. Fountain, Adequate semigroups, Proc. Edinburgh Math. Soc. 22 (2) (1979), 113-125.
  8. J.A. Green, On the structure of semigroups, Ann. of Math. 54 (2) (1951), 163-172.
  9. X.J. Guo and K.P. Shum, On left cyber groups, Intern. Math. Journal 5 (8) (2004), 705-717.
  10. X.J. Guo, Y.Q. Guo and K.P. Shum, Left Abundant semigroups, Communications in Algebras 32 (6) (2004), 2061-2085.
  11. X.J. Guo and K.P. Shum, A structure theorem for perfect abundant semigroups, Asian-Eur. J. Math. 1 (1) (2008), 69-76.
  12. Y.Q. Guo, K.P. Shum and P.Y. Zhu, The structure of left C- rpp semigroups, Semigroup Forum 50 (1995), 9-23.
  13. Y.Q. Guo, The right dual of left C- rpp semigroups, Chinese Sci. Bull. 42 (19) (1997), 1599-1603.
  14. Y.Q. Guo, K.P. Shum and C.M. Gong, On (*,~)-Green's relations and Ortho-lc-monoids, to appear in Communications in Algebras.
  15. L.Du, Y.Q. Guo and K.P. Shum, Some remarks on (l)-Green's relations and strongly rpp semigroups, submitted to Acta Mathematica Scientia 2009.
  16. J.M. Howie, Fundamentals of semigroup theory, London Mathematical Society Monographs, New Series, 12. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York 1995.
  17. X.Z. Kong, On the construction of regular orthocryptogroups, Acta Mathematica Sinica 18 (3) (2002), 517-530.
  18. X.Z. Kong and K.P. Shum, On the structure of regular crypto semigroups, Communications in Algebra 29 (6) (2001), 2461-2479.
  19. X.Z. Kong and K.P. Shum, Completely regular semigroups with generalized strong semilattice decomposition, Algebra Colloqium 12 (2) (2005), 269-280.
  20. X.Z. Kong and K.P. Shum, Normal H̃-abundant cryptogroups, Publ. Math. Debrecen 72 (3-4) (2008), 335-346.
  21. F. Pastijn and M. Petrich, Congruences on regular semigroups, Trans. Amer. Math. Soc. 295 (2) (1986), 607-633.
  22. F. Pastijn, A generalization of Green's equivalence relations for halfgroupoids, Simon Stevin 49 (4) (1975/76), 165-175.
  23. F. Pastijn and M. Petrich, Congruences on regular semigroups associated with Green's relations, Boll. Un. Mat. Ital. B 7 1 (2) (1987), 591-603.
  24. M. Petrich and N.R. Reilly, Completely Regular Semigroups, John Wiley & Sons 1999.
  25. G. Preston, A.H. Clifford's work on unions of groups, Semigroup theory and its applications, New Orleans, LA, (1994), 5-14, London Math. Soc. Lecture Note Ser., 231, Cambridge Univ. Press, Cambridge 1996.
  26. X.M. Ren and K.P. Shum, The structure of superabundant semigroups, Sci. China Ser. A 47 (5) (2004), 756-771.
  27. X.M. Ren and K.P. Shum, The structure of L*-inverse semigroups, Sci. China Ser. A 49 (8) (2006), 1065-1081.
  28. X.M. Ren and K.P. Shum, The structure of Quasi*-inverse semigroups, to appear in Journal of Algebra.
  29. K.P. Shum and Y.Q. Guo, Regular semigroups and their generalizations, Rings, groups, and algebras, 181-226, Lecture Notes in Pure and Appl. Math., 181, Dekker, New York, 1996.
  30. K.P. Shum and X.M. Ren, Abundant semigroups and their special subclasses. Proceedings of the International Conference on Algebra and its Applications (ICAA 2002) (Bangkok), 66-86, Chulalongkorn Univ., Bangkok 2002.
  31. K.P. Shum, X.J. Guo and X.M. Ren, (l)-Green's relations and perfect rpp semigroups, Proceedings of the Third Asian Mathematical Conference, 2000 (Diliman), 604-613, World Sci. Publ., River Edge, NJ, 2002.
  32. R. Zhang, On the refined semilattice decomosition structure of left C-rpp semigroups, Southeast Asian Bull Math. 24 (1) (2000), 137-145.
  33. M. Yamada, Orthodox semigroups whose idempotents satisfy a certain identity, Semigroup Forum 6 (2) (1973), 113-128.
Discussiones Mathematicae - General Algebra and Applications
2010/30/1
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Pages:
71-89
Main language of publication
English
Received
2009-05-06
Accepted
2009-06-22
Published
2010

DOI: 10.7151/dmgaa.1163

Exact and natural sciences