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.

H. Yuan and X. Kong, "Applications of Homomorphism on the Structure of Semigroups,"

References

[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.

http://dx.doi.org/10.2307/1968781

[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. http://dx.doi.org/10.1080/00927870903428247

[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. http://dx.doi.org/10.1090/S0002-9947-1974-0330331-4

[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.

http://dx.doi.org/10.1112/plms/s3-44.1.103

[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. http://dx.doi.org/10.1007/s11425-009-0150-3

[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. http://dx.doi.org/10.1007/s00233-006-0614-5

[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. http://dx.doi.org/10.1007/s00233-006-0614-5

[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. http://dx.doi.org/10.7151/dmgaa.1163

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

http://dx.doi.org/10.1080/00927879708825931

[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. http://dx.doi.org/10.1142/S0218196799000412

[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.

http://dx.doi.org/10.2307/1968781

[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. http://dx.doi.org/10.1080/00927870903428247

[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. http://dx.doi.org/10.1090/S0002-9947-1974-0330331-4

[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.

http://dx.doi.org/10.1112/plms/s3-44.1.103

[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. http://dx.doi.org/10.1007/s11425-009-0150-3

[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. http://dx.doi.org/10.1007/s00233-006-0614-5

[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. http://dx.doi.org/10.1007/s00233-006-0614-5

[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. http://dx.doi.org/10.7151/dmgaa.1163

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

http://dx.doi.org/10.1080/00927879708825931

[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. http://dx.doi.org/10.1142/S0218196799000412