A categorical equivalence for monadic algebras of first-order substructural logics motivated by Kalman's construction

TL;DR

This paper establishes a categorical equivalence between monadic residuated distributive lattices and certain c-differential residuated lattices, extending previous algebraic frameworks for substructural logics.

Contribution

It generalizes existing equivalences to monadic structures and introduces new algebraic properties, addressing limitations of prior work.

Findings

Proves a categorical equivalence between $ ext{MRDL}$ and $ ext{MDRDL'}$.

Introduces the concept of monadic residuated lattices and their properties.

Extends and generalizes previous algebraic logic results.

Abstract

The category $\mathbb{DRDL'}$, whose objects are c-differential residuated distributive lattices that satisfy the condition $\mathbf{CK}$, is the image of the category $\mathbb{RDL}$, whose objects are residuated distributive lattices, under the categorical equivalence (Kalman functor) $\mathbf{K}$. The main goal of this paper is to lift this equivalence $\mathbf{K}$ to the category $\mathbb{MRDL}$, whose objects are monadic residuated distributive lattices, and the category $\mathbb{MDRDL'}$, whose objects are pairs formed by an object of $\mathbb{DRDL'}$ and a center universal quantifier. Firstly, based on the variety of monadic FL$_\textrm{e}$-algebras, we introduce the concept of monadic residuated lattices and study some of their further algebraic properties, proving the classes of monadic residuated distributive lattices and monadic c-differential residuated distributive lattices are in one-to-one correspondence. Subsequently, based on this corresponding relation, we prove that there exists a categorical equivalence between the categories $\mathbb{MRDL}$ and $\mathbb{MDRDL'}$. The results of this paper not only generalizes the works of Sagastume and San Mart\'{i}n in [Mathematical Logic Quarterly, {\bf 60}(2014), 375--388], but also addresses and overcomes the limitations identified in the works of [Studia Logica, {\bf 111}(2023), 361--390]. Finally, this paper concludes with some applications regarding descriptions of a 2-contextual translation.