Additively orthodox semirings with special transversals

Kaiqing Huang; Yizhi Chen; Miaomiao Ren; Kaiqing Huang; Yizhi Chen; Miaomiao Ren

doi:10.3934/math.2022230

AIMS Mathematics

2022, Volume 7, Issue 3: 4153-4167. doi: 10.3934/math.2022230

Previous Article Next Article

Research article Special Issues

Additively orthodox semirings with special transversals

1.
Modern Industrial Innovation Practice Center, Dongguan Polytechnic, Dongguan 523808, China
2.
School of Mathematical Sciences, South China Normal University, Guangzhou 510631, China
3.
Department of Mathematics and Statistics, Huizhou University, Huizhou 516007, China
4.
School of Mathematics, Northwest University, Xi'an 710127, China

Received: 15 August 2021 Revised: 05 December 2021 Accepted: 09 December 2021 Published: 16 December 2021
MSC : 16Y60, 20M10

A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.
- additively orthodox semiring,
- skew-ring transversal,
- generalized Clifford semiring transversal
Citation: Kaiqing Huang, Yizhi Chen, Miaomiao Ren. Additively orthodox semirings with special transversals[J]. AIMS Mathematics, 2022, 7(3): 4153-4167. doi: 10.3934/math.2022230

Related Papers:

Abstract

A semiring $ (S, +, \cdot) $ is called additively orthodox semiring if its additive reduct $ (S, +) $ is a orthodox semigroup. In this paper, by introducing some special semiring transversals as the tools, the constructions of additively orthodox semirings with a skew-ring transversal or with a generalized Clifford semiring transversal are established. Meanwhile, it is shown that an additively orthodox semiring with a generalized Clifford semiring transversal is a b-lattice of additively orthodox semirings with skew-ring transversals. Consequently, the corresponding results of Clifford semirings and generalized Clifford semirings in reference (M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444) and completely regular semirings in reference (S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328) are extended and strengthened.

References

[1]	T. S. Blyth, Lattices and ordered algebraic structures, 1 Eds., London: Springer, 2005. https://doi.org/10.1007/b139095
[2]	T. S. Blyth, M. H. Almeida Santos, On quasi-orthodox semigroups with inverse transversals, P. Edinb. Math. Soc., 40 (1997), 505–514. https://doi.org/10.1017/S0013091500023981 doi: 10.1017/S0013091500023981
[3]	T. S. Blyth, J. F. Chen, Inverse transversals are mutually isomorphic, Comm. Algebra, 29 (2001), 799–804. https://doi.org/10.1081/AGB-100001544 doi: 10.1081/AGB-100001544
[4]	T. S. Blyth, R. McFadden, Regular Semigroups with a multiplicative inverse transversal, P. Roy. Soc. Edinb. A, 92 (1982), 253–270. https://doi.org/10.1017/S0308210500032522 doi: 10.1017/S0308210500032522
[5]	J. F. Chen, On regular semigroups with orthodox transversals, Commun. Algebra, 27 (1999), 4275–4288. https://doi.org/10.1080/00927879908826695 doi: 10.1080/00927879908826695
[6]	A. El-Qallali, Abundant semigroups with a multiplicative type A transversal, Semigroup Forum, 47 (1993), 327–340. https://doi.org/10.1007/BF02573770 doi: 10.1007/BF02573770
[7]	M. P. Grillet, Green's relations in a semiring, Port. Math., 29 (1970), 181–195.
[8]	M. P. Grillet, Semirings with a completely simple additive semigroup, J. Aust. Math. Soc., 20 (1975), 257–267. https://doi.org/10.1017/S1446788700020607 doi: 10.1017/S1446788700020607
[9]	X. J. Guo, Abundant semigroups with a multiplicative adequate transversal, Acta Math. Sin., 18 (2002), 229–244. https://doi.org/10.1007/s101140200170 doi: 10.1007/s101140200170
[10]	S. Ghosh, A characterization of semirings which are subdirect products of a distributive lattice and a ring, Semigroup Forum, 59 (1999), 106–120. https://doi.org/10.1007/PL00005999 doi: 10.1007/PL00005999
[11]	J. S. Golan, Semirings and their applications, 1 Eds., Netherlands: Springer, 1999.
[12]	J. M. Howie, Fundamentals of semigroup theory, 1 Eds., Oxford: Oxford University Press, 1995.
[13]	P. H. Karvellas, Inversive semirings, J. Aust. Math. Soc., 18 (1974), 277–288. https://doi.org/10.1017/S1446788700022850 doi: 10.1017/S1446788700022850
[14]	S. K. Maity, M. K. Sen, K. P. Shum, On completely regular semirings, Bull. Cal. Math. Soc., 98 (2006), 319–328.
[15]	S. K. Maity, R. Ghosh, On quasi completely regular semirings, Semigroup Forum, 89 (2014), 422–430. https://doi.org/10.1007/s00233-014-9579-y doi: 10.1007/s00233-014-9579-y
[16]	D. B. McAlister, R. McFadden, Regular semigroups with inverse transversals, Q. J. Math., 34 (1983), 459–474. https://doi.org/10.1093/qmath/34.4.459 doi: 10.1093/qmath/34.4.459
[17]	D. B. McAlister, R. McFadden, Semigroups with inverse transversal as matrix semigroups, Q. J. Math., 35 (1984), 455–474. https://doi.org/10.1093/qmath/35.4.455 doi: 10.1093/qmath/35.4.455
[18]	F. Pastijn, Y. Q. Guo, Semirings which are unions of rings, Sci. China Ser. A, 45 (2002), 172–195. https://doi.org/10.1360/02ys9020 doi: 10.1360/02ys9020
[19]	T. Saito, Structure of regular semigroup with a quasi-ideal inverse transversal, Semigroup Forum, 31 (1985), 305–309. https://doi.org/10.1007/BF02572659 doi: 10.1007/BF02572659
[20]	T. Saito, Construction of regular semigroups with inverse transversals, P. Edinb. Math. Soc., 32 (1989), 41–51. https://doi.org/10.1017/S0013091500006891 doi: 10.1017/S0013091500006891
[21]	M. K. Sen, Y. Q. Guo, K. P. Shum, A class of idempotent semirings, Semigroup Forum, 60 (2000), 351–367. https://doi.org/10.1007/s002339910029 doi: 10.1007/s002339910029
[22]	M. K. Sen, S. K. Maity, K. P. Shum, Clifford semirings and generalized Clifford semirings, Taiwan. J. Math., 9 (2005), 433–444.
[23]	M. K. Sen, A. K. Bhuniya, Recent Developments of Semirings, Conference: International Conference on Algebra, 2010. https://doi.org/10.1142/9789814366311_0047
[24]	X. L. Tang, Regular semigroups with inverse transversals, Semigroup Forum, 55 (1997), 24–32. https://doi.org/10.1007/PL00005909 doi: 10.1007/PL00005909
[25]	J. Zeleznekow, Regular semirings, Semigroup Forum, 23 (1981), 119–136. https://doi.org/10.1007/BF02676640

Reader Comments

Your name:*

Email:*
© 2022 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)