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On some sets of identities satisfied in Abelian groups

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Gajewska-Kurdziel K. , Mruczek K.

Demonstratio Mathematica

2002

tom Vol. 35, nr 3

447-453

The equational theories were studied in many works (see [4], [5], [6], [7]). Let T be a type of Abelian groups. In this paper we consider the extentions of the equational theory Ex(Gn) defined by so called externally compatible identities of Abelian groups and the identity xn = yn. The equational base of this theory was found in [3]. We prove that each equational theory Cn(Ex^(Gn) U {(phi=epsilon}), where phi=epsilon is an identity of type r, is equal to the extension of the equational theory Cn{Ex{Gn) U E), where E is a finite set of one variable identities of type r. The notation in this paper are the same as in [1].

The finite basis problem for Kiselman monoids

51%

Ashikhmin D. N. , Volkov M. V. , Zhang W. T.

Demonstratio Mathematica

2015

tom Vol. 48, nr 4

475--492

In an earlier paper, the second-named author has described the identities holding in the so-called Catalan monoids. Here we extend this description to a certain family of Hecke–Kiselman monoids including the Kiselman monoids Kn. As a consequence, we conclude that the identities of Kn are nonfinitely based for every n≥4 and exhibit a finite identity basis for the identities of each of the monoids K2 and K3.