APM  Vol.4 No.2 , February 2014
Applications of Homomorphism on the Structure of Semigroups
Abstract: By utilizing homomorphisms and -strong semilattice of semigroups, we show that the Green (*,~)-relation H*,~ is a regular band congruence on a r-ample semigroup if and only if it is a G-strong semilattice of completely J*,~-simple semigroups. The result generalizes Petrich’s result on completely regular semigroups with Green’s relation H a normal band congruence or a regular band congruence from the round of regular semigroups to the round of r-ample semigroups.
Cite this paper: H. Yuan and X. Kong, "Applications of Homomorphism on the Structure of Semigroups," Advances in Pure Mathematics, Vol. 4 No. 2, 2014, pp. 62-70. doi: 10.4236/apm.2014.42010.

[1]   A. H. Clifford and G. B. Preston, “The Algebraic Theory of Semigroups,” American Mathematical Society, New York, 1967, pp. 98-120.

[2]   A. H. Clifford, “Semigroups Admitting Relative Inverses,” Annals of Mathematics, Vol. 42, 1941, pp. 1037-1049.

[3]   Y. Q. Guo, K. P. Shum and C. M. Gong, “On -Green’s Relations and Ortho-Lc-Monoids,” Communications in Algebra, Vol. 39, No. 1, 2011, pp. 5-31.

[4]   J. M. Howie, “Fundamental of Semigroup Theory,” Clarendon Press, Oxford, 1995, pp. 56-73.

[5]   M. Petrich and N. R. Reilly, “Completely Regular Semigroups,” John Wiley & Sons, New York, 1999, pp. 162-197.

[6]   M. Petrich, “The Structure of Completely Regular Semigroups,” Transactions of the American Mathematical Society, Vol. 189, 1974, pp. 211-236.

[7]   M. Petrich, “Lectures in Semigroups,” Wiley & Sons Inc., London, 1976, pp. 124-156.

[8]   F. Pastijn, “A Representation of a Semigroup by a Semigroup of Matrices over a Group with Zero,” Semigroup Forum, Vol. 10, 1975, pp. 238-249.

[9]   J. B. Fountain, “Abundant Semigroups,” Proceedings London Mathematical Society, Vol. 43, No. 3, 1982, pp. 103-129.

[10]   M. V. Lawson, “Rees Matrix Semigroups,” Proceedings of the Edinburgh Mathematical Society, Vol. 33, 1990, pp. 23-39.

[11]   X. M. Ren, K. P. Shum and Y. Q. Guo, “A Generalized Clifford Theorem of Semigroups,” Science China Mathematics, Vol. 53, No. 4, 2010, pp. 1097-1101.

[12]   X. Z. Kong and K. P. Shum, “On the Structure of Regular Crypto Semigroups,” Communications in Algebra, Vol. 29, No. 6, 2001, pp. 2461-2479.

[13]   X. Z. Kong and Z. L. Yuan, “Normal Crypto -Abundant Semigroups,” Advances in Mathematics (in Chinese), Vol. 36, No. 5, 2007, pp. 539-545.

[14]   X. Z. Kong and Z. L. Yuan, “KG-Strong Semilattice Structure of Regular Orthocryptosemigroups,” Semigroup Fourm, Vol. 73, No. 1, 2006, pp. 95-108.

[15]   X. Z. Kong and K. P. Shum, “A Structure Theorem of Normal -Cryptographs,” Publicationes Mathematicae, Vol. 72, No. 3-4, 2008, pp. 335-346.

[16]   X. Z. Kong, Z. L. Yuan and K. P. Shum, “ -Abundant Semigroups and -Cryptographs,” Algebra Collections, Vol. 15, No. 4, 2008, pp. 653-666.

[17]   K. P. Shum, L. Du and Y. Q. Guo, “Green’s Relations and Their Generalizations on Semigroups,” Discussiones Mathematicae —General Algebra and Applications, Vol. 30, No. 1, 2010, pp. 71-89.

[18]   X. D. Tang, “On a Theorem of C-Wrpp Semigroups,” Communications in Algebra, Vol. 25, No. 5, 1997, pp. 1499-1504.

[19]   G. M. S. Gomes and V. Gould, “Proper Weakly Left Ample Semigroups,” International Journal of Algebra and Computation, Vol. 9, No. 6, 1999, pp. 721-739.