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Hopfian and co-hopfian subsemigroups and extensions

Cain A. J., Maltcev V.

Demonstratio Mathematica

2014

Vol. 47, nr 4

791--804

This paper investigates the preservation of hopficity and co-hopficity on passing to finite-index subsemigroups and extensions. It was already known that hopficity is not preserved on passing to finite Rees index subsemigroups, even in the finitely generated case. We give a stronger example to show that it is not preserved even in the finitely presented case. It was also known that hopficity is not preserved in general on passing to finite Rees index extensions, but that it is preserved in the finitely generated case. We show that, in contrast, hopficity is not preserved on passing to finite Green index extensions, even within the class of finitely presented semigroups. Turning to co-hopficity, we prove that within the class of finitely generated semigroups, co-hopficity is preserved on passing to finite Rees index extensions, but is not preserved on passing to finite Rees index subsemigroups, even in the finitely presented case. Finally, by linking co-hopficity for graphs to co-hopficity for semigroups, we show that without the hypothesis of finite generation, co-hopficity is not preserved on passing to finite Rees index extensions.

On subsemigroups of the L1/s

Jabłoński W.

Demonstratio Mathematica

2006

Vol. 39, nr 4

815-823

In this paper we consider subsemigroups of the group L1/s such that the r–th parameter of xs is the function of the remaining ones. Moreover we generalize some results concerning the existence of certain form subsemigroups of the group L1/s.

On some subsemigroups of the groups L1/s and L1/nieskończoność

Jabłoński W.

Demonstratio Mathematica

2006

Vol. 39, nr 2

305-316

The problem of determination of some class of geometric objects has been reduced about forty years ago to consideration of some subsemigroups of the differential group L1/s (cf. [1] and [6]). Over the last years many papers has been devoted the problem of determining of subsemigroups and subgroups of the group L/s (see among others (2], (3] and (5]-[15]). In this paper we are going to generalize the results from [5], [9] and (13] concerning determination of some form subsemigroups of the group L1/s.

On subclasses of groups without free subsemigroups

Bajorska B., Macedońska O.

Demonstratio Mathematica

2000

Vol. 33, nr 1

35-42

The paper is inspired by the question of A. Shalev about possible coincidence of the class of collapsing groups and groups satisfying positive laws. We split the class of collapsing groups for subclasses, corresponding to different functions on natural numbers and give a positive answer for some of them.