Equations on the semidirect product of a finite semilattice by a finite commutative monoid. (English) Zbl 0816.20052

The classes of semigroups studied in this paper are uniformly periodic pseudovarieties of finite commutative monoids. The authors consider the pseudovarieties obtained as semidirect products of each such class with the pseudovariety of all finite semilattices. The resulting product pseudovarieties are characterized in terms of the syntactic monoids which belong to them. Finally the authors obtain a defining set of identities for each such product pseudovariety.


20M07 Varieties and pseudovarieties of semigroups
20M05 Free semigroups, generators and relations, word problems
08C15 Quasivarieties
20M14 Commutative semigroups
Full Text: DOI EuDML


[1] Almeida, J.,On iterated semidirect products of finite semilattices, J. Algebra142 (1991), 239–254. · Zbl 0743.20056 · doi:10.1016/0021-8693(91)90228-Z
[2] Ash, C. J.,Finite semigroups with commuting idempotents, J. Austral. Math. Soc., Ser. A43 (1987), 81–90. · Zbl 0634.20032 · doi:10.1017/S1446788700028998
[3] Blanchet-Sadri, F.,Equations and dot-depth one, Semigroup Forum47 (1993), 305–317. · Zbl 0814.20048 · doi:10.1007/BF02573768
[4] Brzozowski, J. A. and I. Simon,Characterizations of locally testable events, Discrete Math.4 (1973), 243–271. · Zbl 0255.94032 · doi:10.1016/S0012-365X(73)80005-6
[5] Eilenberg, S.,Automata, Languages and Machines, Vol. B, Academic Press, New York, 1976. · Zbl 0359.94067
[6] Eilenberg, S. and M.-P. Schützenberger,On Pseudovarieties, Adv. Math.19 (1976), 413–418. · Zbl 0351.20035 · doi:10.1016/0001-8708(76)90029-3
[7] Irastorza, C.,Base non finie de variétés, Lecture Notes in Computer Science, Springer, Berlin8 (1985), 180–186. · Zbl 0572.20041 · doi:10.1007/BFb0024007
[8] Lallement, G.,Semigroups and Combinatorial Applications, Wiley, New York, 1979. · Zbl 0421.20025
[9] Neumann, H.,Varieties of Groups, Springer, Berlin, 1967. · Zbl 0149.26704
[10] Pin, J.-E.,Variétés de Langages Formels, Masson, Paris, 1984; Varieties of Formal Languages, North Oxford Academic, London 1986 and Plenum, New York, 1986.
[11] Pin, J.-E.,Hiérarchies de concaténation, RAIRO Inform. Théor.18 (1984), 23–46.
[12] Pin, J.-E.,On semidirect products of two finite semilattices, Semigroup Forum28 (1984), 73–81. · Zbl 0527.20046 · doi:10.1007/BF02572474
[13] Ross, K. A. and C.R.B. Wright,Discrete Mathematics, Prentice-Hall, N.J., 1992. · Zbl 0743.00007
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.