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Etale spaces of residuated lattices

Saeed Rasouli, Seyed Naser Hosseini, Amin Dehghani

Abstract

This paper explores the concept of étalé spaces associated with residuated lattices. Notions of bundles and étalés of residuated lattices over a given topological space are introduced and investigated. For a topological space $\mathscr{B}$, we establish that the category of étalés of residuated lattices over $\mathscr{B}$ with morphisms of étalés of residuated lattices is coreflective in the category of bundles of residuated lattices over $\mathscr{B}$ along with morphisms of bundles of residuated lattices. We provide a method for transferring an étalé of residuated lattices over a topological space to another, utilizing a continuous map. Finally, we define a contravariant functor, called the section functor, from the category of étalés of residuated lattices with inverse morphisms to the category of residuated lattices.

Etale spaces of residuated lattices

Abstract

This paper explores the concept of étalé spaces associated with residuated lattices. Notions of bundles and étalés of residuated lattices over a given topological space are introduced and investigated. For a topological space , we establish that the category of étalés of residuated lattices over with morphisms of étalés of residuated lattices is coreflective in the category of bundles of residuated lattices over along with morphisms of bundles of residuated lattices. We provide a method for transferring an étalé of residuated lattices over a topological space to another, utilizing a continuous map. Finally, we define a contravariant functor, called the section functor, from the category of étalés of residuated lattices with inverse morphisms to the category of residuated lattices.
Paper Structure (5 sections, 37 theorems, 41 equations, 11 figures, 6 tables)

This paper contains 5 sections, 37 theorems, 41 equations, 11 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Residuated lattices
  3. Étalé spaces of residuated lattices over a topological space
  4. Interrelation between bundles and étalés of residuated lattices
  5. Étalés of residuated lattices

Key Result

Lemma 3.2

A local homeomorphism is continuous.

Figures (11)

  • Figure 1: Hasse diagram of $\mathfrak{A}_{4}$
  • Figure 2: Hasse diagram of $\mathfrak{A}_{6}$
  • Figure 3: Hasse diagram of $\mathfrak{A}_{8}$
  • Figure 4: Morphism $h$ of the slice category $\mathcal{C}/c$
  • Figure 5: The composition of two $\textbf{RLE}_{inv}$-morphisms
  • ...and 6 more figures

Theorems & Definitions (91)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Example 2.4
  • Example 2.5
  • Example 2.6
  • Example 2.7
  • Example 2.8
  • Definition 3.1
  • Lemma 3.2
  • ...and 81 more