# Semigroups generated by partitions

## TL;DR

This paper investigates the algebraic structure of semigroups generated by specific partitions of the Cartesian square of a set, focusing on the finest and symmetric partitions, revealing their algebraic properties.

## Contribution

It provides a detailed description of the algebraic structure of semigroups generated by the finest and symmetric partitions of $X^2$, a novel analysis in this area.

## Key findings

- Semigroups generated by the finest partition have a specific algebraic structure.
- Semigroups generated by the finest symmetric partition are characterized explicitly.
- The study advances understanding of relation-based semigroup structures.

## Abstract

Let $X$ be a nonempty set and $X^{2}$ be the Cartesian square of $X$. Some semigroups of binary relations generated partitions of $X^2$ are studied. In particular, the algebraic structure of semigroups generated by the finest partition of $X^{2}$ and, respectively, by the finest symmetric partition of $X^{2}$ are described.