Taiwanese Journal of Mathematics

CLIFFORD SEMIRINGS AND GENERALIZED CLIFFORD SEMIRINGS

M. K. Sen, S. K. Maity, and K. P. Shum

Full-text: Open access

Abstract

It is well known that a semigroup $S$ is a Clifford semigroup if and only if $S$ is a strong semilattice of groups. In this paper, we extend this important result from semigroups to semirings by showing that a semiring $S$ is a Clifford semiring if and only if $S$ is a strong distributive lattice of skew-rings. Also, as a further generalization, we prove that a semiring $S$ is a genneralized Clifford semiring if and only if $S$ is a strong b-lattice of skew-rings. Some results which have been recently obtained in the literature [2] are strengthened and extended.

Article information

Source
Taiwanese J. Math., Volume 9, Number 3 (2005), 433-444.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500407851

Digital Object Identifier
doi:10.11650/twjm/1500407851

Mathematical Reviews number (MathSciNet)
MR2162888

Zentralblatt MATH identifier
1091.16028

Subjects
Primary: 16A78 20M07: Varieties and pseudovarieties of semigroups

Keywords
completely regular semiring Clifford semiring generalized Clifford semiring skew-ring b-lattice

Citation

Sen, M. K.; Maity, S. K.; Shum, K. P. CLIFFORD SEMIRINGS AND GENERALIZED CLIFFORD SEMIRINGS. Taiwanese J. Math. 9 (2005), no. 3, 433--444. doi:10.11650/twjm/1500407851. https://projecteuclid.org/euclid.twjm/1500407851


Export citation

References

  • [1.] H. J. Bandelt, and M. Petrich, Subdirect product of rings and distributive lattices. Proc. Edinburgh. Math., 25 (1982), 155-171.
  • [2.] Shamik Ghosh, A characterization of semirings which are subdirect products of a distributive lattice and a ring. Semigroup Forum. 59 (1999), 106-120.
  • [3.] J. S. Golan, The Theory of Semirings with Applications in Mathematics and Surveys in Pure and Applied Mathematics. 54, Longman Scientific and Technical, 1992.
  • [4.] M. P. Grillet, Semirings with a completely simple additive semigroup. J. Austral. Math. Soc., 20 (Series A) (1975), 257-267.
  • [5.] J. M. Howie, Introduction to the theory of semigroups. Academic Press, 1976.
  • [6.] P. H. Karvellas, Inverse semirings. J. Austral. Math. Soc. 18 (1974), 277-288.
  • [7.] F. Pastijn and Y. Q. Guo, The lattice of idempotent distributive semiring varieties. Science in China, 42(8) (Series A) (1999), 785-804.
  • [8.] M. K. Sen, Y. Q. Guo and K. P. Shum, A class of idempotent semirings. Semigroup Forum, 60 (2000), 351-367.
  • [9.] M. K. Sen, S. K. Maity and K. P. Shum, On Completely Regular Semirings. (Submitted).
  • [10.] M. K. Sen, Shamik Ghosh, and P. Mukhopadhyay, Congruences on inversive semirings. Algebras and Combinatorics, Proceedings ICAC 97 (HK), Springer-Verlag (1999), 391-400.
  • [11.] J. Zeleznekow, Regular semirings. Semigroup Forum, 23 (1981), 119-136.
  • [12.] X. Zhao, K. P. Shum and Y. Q. Guo, L-subvarieties of the variety of idempotent semirings. Algebra Universalis, 46 (2001), 75-96.