Residuated Relational Systems

TL;DR

This paper generalizes residuated posets by replacing the partial order with a binary relation, creating residuated relational systems with properties similar to those of posets and lattices.

Contribution

It introduces a new framework for residuated systems using arbitrary binary relations, extending the classical concept of residuated posets.

Findings

Residuated relational systems exhibit properties analogous to residuated posets.

Enriching the binary relation yields interesting algebraic properties.

The framework broadens the applicability of residuated structures.

Abstract

The aim of the present paper is to generalize the concept of residuated poset, by replacing the usual partial ordering by a generic binary relation, giving rise to relational systems which are residuated. In particular, we modify the definition of adjointness in such a way that the ordering relation can be harmlessly replaced by a binary relation. By enriching such binary relation with additional properties we get interesting properties of residuated relational systems which are analogical to those of residuated posets and lattices.