Some special congruences on completely regular semigroups

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.