×

A non-regular perfect semigroup. (English) Zbl 0615.20041

A congruence \(\theta\) on a semigroup S is perfect if \((a\theta)(b\theta)=(ab)\theta\) for every a,b\(\in S\), and S is perfect when all its congruences are perfect. It is known that all commutative and all finite perfect semigroups are regular. Here it is given an example of a non-regular (cancellative) perfect semigroup.
Reviewer: Ja.Henno

MSC:

20M10 General structure theory for semigroups
20M15 Mappings of semigroups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Clifford, A.H. and G.B. Preson,The algebraic theory of semigroups, vol. I, Amer. Math. Soc., Providence, R.I., 1961.
[2] Fortunatov, V.A.,Perfect semigroups decomposable in a semilattice of rectangular groups, Studies in Algebra2 (1970), 67–78 (Saratov Univ. Press (in Russian)). (MR 53, #8290) · Zbl 0245.20061
[3] Fortunatov, V.A., Perfect semigroups, Izv. Vyss. Ucebn. Zaved. Matematika3 (1972), 80–90 (in Russian). (MR 45, #8752)
[4] Fortunatov, V.A.,Varieties of perfect algebras, Studies in Algebra4 (1974), 110–114 (Saratov Univ. Press (in Russian)). (MR 54, #12603)
[5] Fortunatov, V.A.,Congruences on simple extensions of semigroups, Semigroup Forum13 (1977), 283–295. · Zbl 0354.20049 · doi:10.1007/BF02194949
[6] Hamilton, H.,Perfect completely regular semigroups, Math. Nachr. 123 (1984), 169–176. · Zbl 0576.20039 · doi:10.1002/mana.19851230116
[7] Hamilton, H. and T. Tamura,Finite inverse perfect semigroups and their congruences, J. Austral. Math. Soc. (Series A)32 (1982), 114–128. · Zbl 0486.20038 · doi:10.1017/S1446788700024459
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.