×

Varieties of completely regular semigroups generated by Mal’cev products. (English) Zbl 0808.20050

The Malcev product of two varieties \(\mathcal U\), \(\mathcal V\) of completely regular semigroups consists of those completely regular semigroups \(S\) endowed with a congruence whose idempotent classes belong to \(\mathcal U\) and whose factor semigroup belongs to \(\mathcal V\). The reviewer [J. Aust. Math. Soc., Ser. A 42, 227-246 (1987; Zbl 0613.20038)] showed that if \(\mathcal U\) consists of rectangular groups and \(\mathcal V\) is any variety of completely regular semigroups, then their Malcev product is always a variety and asked if this remains true for all varieties \(\mathcal U\) of completely simple semigroups (all other cases have been decided). The authors first extend the above result to all such varieties \(\mathcal U\) that are central, that is, in each member of \(\mathcal U\) products of idempotents lie in the centre of the maximal subgroup to which they belong. They go on to answer the above question in the negative: the Malcev product of the variety \(\mathcal D\), consisting of those completely simple semigroups for which the idempotent-generated core has abelian maximal subgroups, with the variety of left regular bands, is not again a variety.
In a related study, the varieties generated by various important Malcev products are described, making use of the concept of “CR-relational morphism”, an adaptation of the usual concept of relational morphism in arbitrary semigroups.

MSC:

20M07 Varieties and pseudovarieties of semigroups
20M17 Regular semigroups
08B25 Products, amalgamated products, and other kinds of limits and colimits

Citations:

Zbl 0613.20038
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Clifford, A. H.,The free completely regular semigroup on a set, J. Algebra59 (1979), 434–451. · Zbl 0412.20049 · doi:10.1016/0021-8693(79)90139-X
[2] Eilenberg, S.,Automata, languages and machines, Vol. B, Academic Press, New York, 1976. · Zbl 0359.94067
[3] Grätzer, G.,Universal algebra, 2nd ed., Springer-Verlag, New York, 1979. · Zbl 0412.08001
[4] Howie, J. M.,An introduction to semigroup theory, Academic Press, London, 1976. · Zbl 0355.20056
[5] Jones, P. R.,Mal’cev products of varieties of completely regular semigroups, J. Austral. Math. Soc. (Series A)42 (1987), 227–246. · Zbl 0613.20038 · doi:10.1017/S1446788700028226
[6] Mal’cev, A. I.,The metamathematics of algebraic systems, North-Holland, Amsterdam, 1971.
[7] McAlister, D. B. and N. R. Reilly,E-unitary covers for inverse semigroups, Pacific J. of Math.68 (1977), 161–174. · Zbl 0368.20043
[8] Pastijn, F. and P. G. Trotter,Lattices of completely regular semigroup varieties, Pacific J. of Math.119 (1985), 191–214. · Zbl 0578.20055
[9] Petrich, M.,Introduction to semigroups, Merrill, Columbus, Ohio, 1973. · Zbl 0321.20037
[10] Petrich, M.,The structure of completely regular semigroups, Trans. Amer. Math. Soc.189 (1974), 211–236. · Zbl 0291.20082 · doi:10.1090/S0002-9947-1974-0330331-4
[11] Petrich, M.,Lectures in semigroups, Akademie-Verlag, Berlin, 1977. · Zbl 0369.20036
[12] Petrich, M.,On the varieties of completely regular semigroups, Semigroup Forum25 (1982), 153–169. · Zbl 0502.20034 · doi:10.1007/BF02573595
[13] Petrich, M. and N. R. Reilly,Near varieties of idempotent generated completely simple semigroups, Algebra Universalis16 (1983), 83–104. · Zbl 0516.20037 · doi:10.1007/BF01191755
[14] Petrich, M. and N. R. Reilly,All varieties of central completely simple semigroups, Trans. Amer. Math. Soc.280 (1983), 623–636. · Zbl 0537.20031 · doi:10.1090/S0002-9947-1983-0716841-1
[15] Petrich, M. and N. R. Reilly,Operators related to E-disjunctive and fundamental completely regular semigroups, J. Algebra134 (1990), 1–27. · Zbl 0706.20043 · doi:10.1016/0021-8693(90)90207-5
[16] Pin, J. E.,Varieties of formal languages, North Oxford Academic Publisher Ltd., London, 1986. · Zbl 0632.68069
[17] Pin, J. E.,On a conjecture of Rhodes, Semigroup Forum30 (1989), 1–15. · Zbl 0673.20034 · doi:10.1007/BF02573280
[18] Reilly, N. R.,Varieties of completely regular semigroups, J. Austral. Math. Soc. (Series A)38 (1985), 372–393. · Zbl 0572.20040 · doi:10.1017/S144678870002365X
[19] Reilly, N. R.,Free objects in certain varieties of semigroups, in ”2o Seminario di Algebra non Commutativa”, Siena, 1987, 39–99.
[20] Reilly, N. R.,Structure, congruences and varieties of completely regular semigroups, in ”Seminario di Algebra non Commutativa”, Lecce, 1989, 55–85.
[21] Tamura, T.,Decompositions of a completely simple semigroup, Osaka Math. J.12 (1960), 269–275. · Zbl 0096.01103
[22] Tilson, B.,Categories as algebra: an essential ingredient in the theory of monoids, J. Pure and Appl. Algebra48 (1987), 83–198. · Zbl 0627.20031 · doi:10.1016/0022-4049(87)90108-3
[23] Trotter, P. G.,The least orthodox congruence on a completely regular semigroup, Indian J. Math.25 (1983), 125–133. · Zbl 0577.20057
[24] Zhang, S.,Completely regular semigroup varieties generated by Mal’cev products with groups, to appear in Semigroup Forum. · Zbl 0797.20048
[25] Zhang, S.,Certain operators related to Mal’cev products on varieties of completely regular semigroups, to appear in J. of Algebra. · Zbl 0815.20050
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.