CONGRUENCES ON ABUNDANT SEMIGROUPS WITH QUASI-IDEAL S-ADEQUATE TRANSVERSALS

Wang, Lili;Wang, Aifa

• Received : 2012.06.07
• Published : 2014.01.31
• 29 6

Abstract

In this paper, we give congruences on an abundant semigroup with a quasi-ideal S-adequate transversal $S^{\circ}$ by the congruence pair abstractly which consists of congruences on the structure component parts R and ${\Lambda}$. We prove that the set of all congruences on this kind of semigroups is a complete lattice.

Keywords

abundant semigroup;adequate transversal;quasi-ideal transversal;congruence;congruence pair

References

1. T. S. Blyth and R. B. McFadden, Regular semigroups with a multiplicative inverse transversal, Proc. Roy. Soc. Edinburgh Sect. A 92 (1982), no. 3-4, 253-270. https://doi.org/10.1017/S0308210500032522
2. J. F. Chen, Abundant semigroups with adequate transversals, Semigroup Forum 60 (2000), no. 1, 67-79. https://doi.org/10.1007/s002330010004
3. A. El-Qallali, Abundant semigroups with a multiplicative type A transversal, Semigroup Forum 47 (1993), no. 3, 327-340. https://doi.org/10.1007/BF02573770
4. J. B. Fountain, Adequate semigroups, Proc. Edinburgh Math. Soc. 22 (1979), no. 2, 113-125. https://doi.org/10.1017/S0013091500016230
5. J. B. Fountain, Abundant semigroups, Proc London Math. Soc. 44 (1982), no. 1, 103-129.
6. X. J. Guo, Abundant semigroups with a multiplicative adequate transversal, Acta Math. Sin 18 (2002), no. 2, 229-244. https://doi.org/10.1007/s101140200170
7. X. J. Guo and L. M. Wang, Idempotent-connected abundant semigroups which are disjoint unions of quasi-ideal adequate transversals, Comm. Algebra 30 (2002), no. 4, 1779-1800. https://doi.org/10.1081/AGB-120013215
8. X. J. Kong and P. Wang, Abundant semigroups with quasi-ideal S-adequate transversals, Commun. Korean Math. Soc. 26 (2011), no. 1, 1-12. https://doi.org/10.4134/CKMS.2011.26.1.001
9. X. L. Tang, Regular semigroups with inverse transversals, Semigroup Forum 55 (1997), no. 1, 24-32. https://doi.org/10.1007/PL00005909
10. X. L. Tang and L. M. Wang, Congruences on regular semigroups with inverse transversals, Comm. Algebra 23 (1995), no. 11, 4157-4171. https://doi.org/10.1080/00927879508825455
11. L. M. Wang, On congruence lattice of regular semigroups with Q-inverse transversals, Semigroup Forum 50 (1995), no. 2, 141-160. https://doi.org/10.1007/BF02573513
12. L. M. Wang and X. L. Tang, Congruence lattices of regular semigroups with inverse transversals, Comm. Algebra 26 (1998), no. 4, 1234-1255.