Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Comaximal filter graphs in residuated lattices

Surdive Atamewoue, Hugue Tchantcho

TL;DR

This work introduces the comaximal filter graph Cf(A) for a commutative, integral, non-degenerate residuated lattice A, with vertices as proper filters not contained in the radical and edges reflecting comaximality via the generated filter equaling the whole lattice. It connects Cf(A) to zero-divisor graphs, showing Cf(A) embeds as a subgraph of the zero-divisor graph and, in the finite case, can be isomorphic under a cardinality condition on zero-divisors; it also examines when lattice isomorphisms induce graph isomorphisms. A central result is that for finite A, the chromatic number and clique number of Cf(A) coincide and equal the number of maximal filters, yielding a natural |Max(A)|-partite structure and guiding coloring strategies. The paper also proves planarity for all Cf(A) with |A| ≤ 10 and provides extensive computational data across several residuated-lattice classes, highlighting that graph isomorphism does not determine lattice isomorphism. Overall, the work deepens the link between residuated-lattice theory and graph invariants, offering both theoretical insights and concrete enumerative results for small algebras.

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.

Comaximal filter graphs in residuated lattices

TL;DR

This work introduces the comaximal filter graph Cf(A) for a commutative, integral, non-degenerate residuated lattice A, with vertices as proper filters not contained in the radical and edges reflecting comaximality via the generated filter equaling the whole lattice. It connects Cf(A) to zero-divisor graphs, showing Cf(A) embeds as a subgraph of the zero-divisor graph and, in the finite case, can be isomorphic under a cardinality condition on zero-divisors; it also examines when lattice isomorphisms induce graph isomorphisms. A central result is that for finite A, the chromatic number and clique number of Cf(A) coincide and equal the number of maximal filters, yielding a natural |Max(A)|-partite structure and guiding coloring strategies. The paper also proves planarity for all Cf(A) with |A| ≤ 10 and provides extensive computational data across several residuated-lattice classes, highlighting that graph isomorphism does not determine lattice isomorphism. Overall, the work deepens the link between residuated-lattice theory and graph invariants, offering both theoretical insights and concrete enumerative results for small algebras.

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.
Paper Structure (7 sections, 42 theorems, 6 equations, 2 tables)

This paper contains 7 sections, 42 theorems, 6 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. On residuated lattices
  4. On graphs theory
  5. The comaximal filter graphs in residuated lattices
  6. More on integer invariants of the graph $\mathcal{C}_f(\mathcal{A})$
  7. Some remarks on graph $\mathcal{C}_f(\mathcal{A})$, where $|\mathcal{A}|\leq 10$

Key Result

Proposition 2.1

T01 Let $\mathcal{A}$ be a residuated lattice. If $\emptyset \neq F \subseteq A$ then, $F$ is a filter of $\mathcal{A}$ if and only if for any $x, y \in A$; $x \in F$ and $x\rightarrow y \in F$ imply $y \in F$. That means $F$ is a deductive system of $\mathcal{A}$.

Theorems & Definitions (69)

  • Definition 1
  • Definition 2
  • Proposition 2.1
  • Proposition 2.2
  • Definition 3
  • Proposition 3.1
  • proof
  • Example 1
  • Proposition 3.2
  • proof
  • ...and 59 more