Some special congruences on completely regular semigroups

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.