Residuated implications derived from quasi-overlap functions on lattices

TL;DR

This paper explores residuated implications derived from quasi-overlap functions on lattices, establishing their properties, continuity conditions, and automorphism-based conjugations within a topological lattice framework.

Contribution

It introduces residuated implications from quasi-overlap functions on lattices and characterizes their properties, including continuity and automorphism conjugation, expanding the theoretical understanding of these functions.

Findings

Residuated implications are derived from quasi-overlap functions on lattices.

The class of quasi-overlap functions satisfying residuation is exactly the continuous functions in Scott topology.

Automorphisms are extended to quasi-overlap functions, enabling conjugation within topological lattice spaces.

Abstract

In this paper, we introduce the concept of residuated implications derived from quasi-overlap functions on lattices and prove some related properties. In addition, we formalized the residuation principle for the case of quasi-overlap functions on lattices and their respective induced implications, as well as revealing that the class of quasi-overlap functions that fulfill the residuation principle is the same class of continuous functions according to topology of Scott. Also, Scott's continuity and the notion of densely ordered posets are used to generalize a classification theorem for residuated quasi-overlap functions. Finally, the concept of automorphisms are extended to the context of quasi-overlap functions over lattices, taking these lattices into account as topological spaces, with a view to obtaining quasi-overlap functions conjugated by the action of automorphisms.