arXiv subjects

Anjeza Krakulli

Publications and source records attributed to Anjeza Krakulli.

3 recordsLinked to original sources

arXiv2025

A groupoid approach to the study of fuzzy topological spaces

The definition of the complement of a fuzzy subset is algebraic in nature and when it is used in the context of fuzzy topological spaces it does not share any similarity with the usual property of topological spaces that the complement of an open subset is closed. To tackle this inconsistency, we associate to any fuzzy topological space a topological space and use its fundamental groupoid equipped with the Lasso topology to give a topological characterization for the complementation of fuzzy subsets.

math.GN↗

arXiv2023

Fuzzy semigroups via semigroups

The theory of fuzzy semigroups is a branch of mathematics that arose in early 90's as an effort to characterize properties of semigroups by the properties of their fuzzy subsystems which include, fuzzy subsemigroups and their alike, fuzzy one (resp. two) sided ideals, fuzzy quasi-ideals, fuzzy bi-ideals etc. To be more precise, a fuzzy subsemigroup of a given semigroup $(S,\cdot)$ is just a $\wedge$-prehomomorphism $f$ of $(S,\cdot)$ to $([0,1],\wedge)$. Variations of this, which correspond to the other before mentioned fuzzy subsystems, can be obtained by imposing certain properties to $f$. It turns out from the work of Kuroki, Mordeson, Malik and that of many of their descendants, that fuzzy subsystems play a similar role to the structure theory of semigroups that play their non fuzzy analogues. The aim of the present paper is to show that this similarity is not coincidental. As a first step to this, we prove that there is a 1-1 correspondence between fuzzy subsemigroups of $S$ and subsemigroups of a certain type of $S\times I$. Restricted to fuzzy one sided ideals, this correspondence identifies the above fuzzy subsystems to their analogues of $S\times I$. Using these identifications, we prove that the characterization of the regularity of semigroups in terms of fuzzy one sided ideals and fuzzy quasi-ideals can be obtained as an implication of the corresponding non fuzzy analogue.

math.GM↗

arXiv2014

The Eilenberg-Mac Lane cohomology of an inverse monoid and the maximum group image

The aim of this paper is to see at what extent homological properties of an inverse monoid are determined from those of its maximum group image. We provide several evidences that the maximum group image contains vital homological information which can be used to study certain properties of the monoid itself. For instance, we prove that an inverse monoid $S$ is of type $FP_{\infty}$, if and only if it contains a minimal idempotent and its maximum group image is of the same type. Regarding cohomological dimensions, we show that the cohomological dimension of a free Clifford monoid and that of its maximum group image agree and are equal to one. Also we define the index of a full submonoid of an inverse monoid in terms of their maximum group images and show that if the index is finite then, the monoid is of type $\text{FP}_{\infty}$ if and only if its submonoid is of the same type.

math.GR↗