# Quasicomplemented residuated lattices

## TL;DR

This paper introduces and studies quasicomplemented residuated lattices, a new subclass characterized by properties of prime filters and their relation to dense elements, expanding the understanding of residuated lattice structures.

## Contribution

It defines quasicomplemented residuated lattices, introduces disjunctive residuated lattices, and provides characterizations using $eta$-filters, advancing lattice theory.

## Key findings

- A residuated lattice is Boolean iff it is disjunctive and quasicomplemented.
- Prime filters not containing dense elements are minimal prime filters.
- Characterizations of quasicomplemented residuated lattices via $eta$-filters.

## Abstract

In this paper, the class of quasicomplemented residuated lattices is introduced and investigated, as a subclass of residuated lattices in which any prime filter not containing any dense element is a minimal prime filter. The notion of disjunctive residuated lattices is introduced and it is observed that a residuated lattice is Boolean if and only if it is disjunctive and quasicomplemented. Finally, some characterizations for quasicomplemented residuated lattices are given by means of the new notion of $\alpha$-filters.