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Pseudogroups of transformations on discrete topological spaces

Lipińska J.

Journal of Applied Mathematics and Computational Mechanics

2016

Vol. 15, nr 2

71--75

In all the earlier papers pseudogroups of transformations, generalised inverse semigroups and their connections were considered. This paper is a kind of recapitulation of these problems. Considering pseudogroups of transformations on discrete topological spaces is a step in the same direction. In this paper we notice that domains of functions belonging to a pseudogroup of transformations on discrete topological spaces - when we join an empty set to them create not only a topological space but also σ-body. We also consider pseudogroups on discrete topological spaces with the finite number of elements. The third problem is the influence of topology on relation of partial order.

Properties of a set of idempotent elements of generalised inverse semigroups

Lipińska J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2012

Vol. 11, nr 3

93-97

In the last paper we considered properties of partial order in a generalized inverse semigroup. In this paper we show that a set of idempotent elements of a generalized inverse semigroup is a commutative generalised inverse semigroup. The next problem which appears is if the only commutative inverse semigroups are these which consist of idempotent elements. We prove that it is not the case. We show the commutative inverse semigroup which do not consist only of idempotent elements. We also demonstrate that this inverse generalised semigroup is isomorphic to Ehresmann's pseudogroup.

Properties of partial order in generalised inverse semigroups

Lipińska J.

Prace Naukowe Instytutu Matematyki i Informatyki Politechniki Częstochowskiej

2011

Vol. 10, nr 1

145-149

In this paper we show that among the idempotent elements of a generalised inverse semigroup isomorphic to a pseudogroup of transformations on a topological space, is the largest element. We also show how to obtain the smallest element in some inverse semigroups.