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

## TL;DR

This paper explores properties of congruence sequences in inverse semigroups, introducing new classes and establishing implications within the min network framework.

## Contribution

It introduces new classes of inverse semigroups and identifies least congruences within these classes, expanding the understanding of congruence networks.

## Key findings

- Identification of least $eta_{n+2}$-congruences on inverse semigroups
- Introduction of $eta_n$-is-over-$E$-unitary semigroups
- New implications for quasivarieties induced by the min network

## 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 $\ker{\alpha_n}$-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.