Comaximal filter graphs in residuated lattices

TL;DR

This paper introduces the comaximal filter graph for residuated lattices, exploring its properties and connections to zero-divisor graphs, and analyzing its chromatic number, clique number, planarity, and perfection.

Contribution

It defines and studies the properties of comaximal filter graphs in residuated lattices, linking them to zero-divisor graphs and analyzing their structural characteristics.

Findings

Established properties of the comaximal filter graph

Linked comaximal filter graphs to zero-divisor graphs

Analyzed graph invariants like chromatic and clique numbers

Abstract

Consider A to be a commutative, integral and non-degenerate residuated lattice. In this work, we introduce the graph of comaximal filters on the residuated lattice A. We willdenote by Cf(A) this graph for which the set of vertices are proper filters of A which are not contained in the radical of A, and the adjacency relation on vertices is given as: consider two filters F and G, there are adjacent if and only if F v G; the filter generated by F u G is equal to A. The elementary properties of this graph are provided, we establish a link between this comaximal filter graphs and the zero-divisor graphs. Furthermore, we investigate the chromatic number, the clique number, the planarity and the perfection of the graph Cf(A). We also briefly describe all the comaximal filter graphs constructed on specific residuated lattices of small size.