A new approach to a network of congruences on an inverse semigroup

Abstract

This paper enriches the list of known properties of congruence sequences starting from the universal relation and successively performing the operators lower $k$ and lower $t$. Two series of inverse semigroups, namely $\ker \hat{I}{\pm }_n$-is-Clifford semigroups and ${\hat{I}}_n^2$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely $\hat{I}{\pm }_{n+2}$ and ${\hat{I}}_{n+2}^2$, are found to be the least $\ker \hat{I}{\pm }_n$-is-Clifford and least ${\hat{I}}_n^2$-is-over-$E$-unitary congruences on $S$, respectively. A new system of implications is established for the quasivarieties of inverse semigroups induced by the min network.