# Relatively residuated lattices and posets

## TL;DR

This paper introduces a broader class of relative residuated lattices that include non-modular cases, extending known properties to these structures and to posets, thus broadening the scope of residuated lattice theory.

## Contribution

It defines a new, more general concept of relative residuated lattices that encompasses non-modular and non-distributive cases, expanding the theoretical framework.

## Key findings

- Derived properties similar to classical residuated lattices
- Extended results to posets
- Included non-modular sectionally pseudocomplemented lattices

## Abstract

It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relative residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relative residuated lattices which are similar to those known for residuated ones and extend our results to posets.