# Green's relations and their generalizations on semigroups

Kar-Ping Shum; Lan Du; Yuqi Guo

Discussiones Mathematicae - General Algebra and Applications (2010)

- Volume: 30, Issue: 1, page 71-89
- ISSN: 1509-9415

## Access Full Article

top## Abstract

top## How to cite

topKar-Ping Shum, Lan Du, and Yuqi Guo. "Green's relations and their generalizations on semigroups." Discussiones Mathematicae - General Algebra and Applications 30.1 (2010): 71-89. <http://eudml.org/doc/276488>.

@article{Kar2010,

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.},

author = {Kar-Ping Shum, Lan Du, Yuqi Guo},

journal = {Discussiones Mathematicae - General Algebra and Applications},

keywords = {Green's relations; generalized Green's relations; regular semigroups; abundant semigroups rpp-semigroups; generalized Green relations; abundant semigroups; rpp-semigroups; completely regular semigroups},

language = {eng},

number = {1},

pages = {71-89},

title = {Green's relations and their generalizations on semigroups},

url = {http://eudml.org/doc/276488},

volume = {30},

year = {2010},

}

TY - JOUR

AU - Kar-Ping Shum

AU - Lan Du

AU - Yuqi Guo

TI - Green's relations and their generalizations on semigroups

JO - Discussiones Mathematicae - General Algebra and Applications

PY - 2010

VL - 30

IS - 1

SP - 71

EP - 89

AB - 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.

LA - eng

KW - Green's relations; generalized Green's relations; regular semigroups; abundant semigroups rpp-semigroups; generalized Green relations; abundant semigroups; rpp-semigroups; completely regular semigroups

UR - http://eudml.org/doc/276488

ER -

## References

top- [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. Zbl0111.03403
- [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. Zbl0178.01203
- [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. Zbl0501.20043
- [4] A. El-Qallali and J.B. Fountain, Quasi-adequate semigroups, Proc. Roy. Soc. Edinburgh Sect. A 91 (1-2) (1981/82), 91-99. Zbl0501.20042
- [5] J.B. Fountain, A class of right PP monoids, Quart. J. Math. Oxford Ser. 2, 28 (111) (1977) 285-300. Zbl0377.20051
- [6] J.B. Fountain, Abundant semigroups, Proc. London Math. Soc. 44 (3) (1982), 103-129. Zbl0481.20036
- [7] J.B. Fountain, Adequate semigroups, Proc. Edinburgh Math. Soc. 22 (2) (1979), 113-125. Zbl0414.20048
- [8] J.A. Green, On the structure of semigroups, Ann. of Math. 54 (2) (1951), 163-172. Zbl0043.25601
- [9] X.J. Guo and K.P. Shum, On left cyber groups, Intern. Math. Journal 5 (8) (2004), 705-717. Zbl1158.20329
- [10] X.J. Guo, Y.Q. Guo and K.P. Shum, Left Abundant semigroups, Communications in Algebras 32 (6) (2004), 2061-2085. Zbl1077.20068
- [11] X.J. Guo and K.P. Shum, A structure theorem for perfect abundant semigroups, Asian-Eur. J. Math. 1 (1) (2008), 69-76. Zbl1146.20039
- [12] Y.Q. Guo, K.P. Shum and P.Y. Zhu, The structure of left C- rpp semigroups, Semigroup Forum 50 (1995), 9-23. Zbl0821.20046
- [13] Y.Q. Guo, The right dual of left C- rpp semigroups, Chinese Sci. Bull. 42 (19) (1997), 1599-1603. Zbl0903.20035
- [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. Zbl1247.20068
- [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. Zbl1249.20055
- [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. Zbl0835.20077
- [17] X.Z. Kong, On the construction of regular orthocryptogroups, Acta Mathematica Sinica 18 (3) (2002), 517-530. Zbl1019.20027
- [18] X.Z. Kong and K.P. Shum, On the structure of regular crypto semigroups, Communications in Algebra 29 (6) (2001), 2461-2479. Zbl0989.20052
- [19] X.Z. Kong and K.P. Shum, Completely regular semigroups with generalized strong semilattice decomposition, Algebra Colloqium 12 (2) (2005), 269-280. Zbl1103.20060
- [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. Zbl0599.20095
- [22] F. Pastijn, A generalization of Green's equivalence relations for halfgroupoids, Simon Stevin 49 (4) (1975/76), 165-175. Zbl0335.20028
- [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. Zbl0621.20036
- [24] M. Petrich and N.R. Reilly, Completely Regular Semigroups, John Wiley & Sons 1999. Zbl0967.20034
- [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. Zbl1078.20057
- [27] X.M. Ren and K.P. Shum, The structure of L*-inverse semigroups, Sci. China Ser. A 49 (8) (2006), 1065-1081. Zbl1117.20043
- [28] X.M. Ren and K.P. Shum, The structure of Quasi*-inverse semigroups, to appear in Journal of Algebra. Zbl1219.20040
- [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. Zbl0855.20051
- [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. Zbl1071.20512
- [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. Zbl1015.20045
- [32] R. Zhang, On the refined semilattice decomosition structure of left C-rpp semigroups, Southeast Asian Bull Math. 24 (1) (2000), 137-145. Zbl0976.20045
- [33] M. Yamada, Orthodox semigroups whose idempotents satisfy a certain identity, Semigroup Forum 6 (2) (1973), 113-128. Zbl0255.20038