Remarks on the paper "M. Kolibiar, On a construction of semigroups"

TL;DR

This paper revises Kolibiar's construction of semigroups, clarifying the correct conditions and addressing inaccuracies in the original claims about deriving semigroups via the onstruction.

Contribution

It provides a corrected and clarified version of Kolibiar's semigroup onstruction, fixing errors and refining the theoretical framework.

Findings

Corrected the original onstruction method.

Clarified the relationship between semigroups and their factor semigroups.

Addressed inaccuracies in Kolibiar's original assertions.

Abstract

In his paper "On a construction of semigroups", M. Kolibiar gives a construction for a semigroup $T$ (beginning from a semigroup $S$) which is said to be derived from the semigroup $S$ by a $\theta$-construction. He asserted that every semigroup $T$ can be derived from the factor semigroup $T/\theta (T)$ by a $\theta$-construction, where $\theta (T)$ is the congruence on $T$ defined by: $(a, b)\in \theta (T)$ if and only if $xa=xb$ for all $x\in T$. Unfortunately, the paper contains some incorrect part. In our present paper we give a revision of the paper.