# $n$-normal residuated lattices

## TL;DR

This paper introduces and studies $n$-normal residuated lattices, characterizing their structure through prime filters, minimal prime filters, coannulets, and $\omega$-filters, and explores the properties of $\omega$-filters forming a distributive lattice.

## Contribution

It defines the concept of $n$-normal residuated lattices and characterizes them using prime filters, minimal prime filters, coannulets, and $\omega$-filters, including the structure of $\omega$-filters.

## Key findings

- The set of $\omega$-filters forms a distributive lattice.
- $n$-normal residuated lattices are characterized by properties of prime and minimal prime filters.
- The class of $n$-normal residuated lattices is fully described in terms of filters and $\omega$-filters.

## Abstract

The notion of $n$-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most $n$ minimal prime filters, is introduced and studied. Before that, the notion of $\omega$-filter is introduced and it is observed that the set of $\omega$-filters in a residuated lattice forms a distributive lattice on its own, which includes the set of coannulets as a sublattice. The class of $n$-normal residuated lattices is characterized in terms of their prime filters, minimal prime filters, coannulets and $\omega$-filters.