# Some special congruences on completely regular semigroups

## TL;DR

This paper investigates specific congruence sequences in completely regular semigroups, identifying least congruences that produce quotient semigroups with particular properties, thus addressing open problems in the field.

## Contribution

It introduces new properties of congruence sequences and characterizes least congruences for classes of completely regular semigroups, solving three open problems.

## Key findings

- Identifies least congruences for semigroups with cryptogroup kernels.
- Characterizes least congruences for semigroups over rectangular bands.
- Provides solutions to three open problems in the literature.

## Abstract

This paper enriches the list of properties of the congruence sequences starting from the universal relation and successively performing the operations of lower $t$ and lower $k$. Three classes of completely regular semigroups, namely semigroups for which $\ker{\sigma}$ is a cryptogroup, semigroups for which $\ker{\nu}$ is a cryptogroup and semigroups for which $\kappa$ is over rectangular bands, are studied. $((\omega_t)_k)_t$, $((\mathcal{D}_t)_k)_t$ and $((\omega_k)_t)_k$ are found to be the least congruences on $S$ such that the quotient semigroups are semigroups for which $\ker{\sigma}$ is a cryptogroup, $\ker{\nu}$ is a cryptogroup and $\kappa$ is over rectangular bands, respectively. The results obtained present a response to three problems in Petrich and Reilly's textbook \textquoteleft\textquoteleft Completely Regular Semigroups\textquoteright\textquoteright.