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.