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 $\mathscr{B}$, 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.