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

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 $\ker {\alpha }_n$-is-Clifford semigroups and ${\beta }_n$-is-over-$E$-unitary semigroups, are investigated. Two congruences, namely ${\alpha }_{n+2}$ and ${\beta }_{n+2}$, are found to be the least $\ker {\alpha }_n$-is-Clifford and least ${\beta }_n$-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.