A categorical equivalence for monadic algebras of first-order substructural logics motivated by Kalman's construction
Juntao Wang, Mei Wang, William Zuluaga Botero
TL;DR
This work extends Kalman’s categorical equivalence from residuated distributive lattices to the monadic setting by formulating and relating $\mathbb{MRDL}$ and $\mathbb{MDRDL'}$, i.e., monadic residuated distributive lattices and monadic c-differential residuated lattices with CK. It develops the theory of monadic residuated lattices, establishes a bijection with monadic c-differential residuated lattices satisfying $\mathbf{CK}$, and achieves a full categorical equivalence between $\mathbb{MRDL}$ and $\mathbb{MDRDL'}$ via the Kalman functor $\mathbf{K}$ and its adjoint $\mathbf{C}$, including explicit natural isomorphisms $\phi$ and $\psi$. The paper also constructs a concrete 2-contextual translation from the equational theory of $\mathbb{MDRDL'}$ into that of $\mathbb{MRDL}$, using a detailed description of $\mathbf{C}(\mathbf{F}_{\mathbb{MDRDL'}}(n))$ as a quotient with duplicated generators and the relation $x_i^{1}\odot x_i^{2}=0$. Together, these results generalize prior work and provide a robust algebraic-categorical framework for monadic fragments of first-order substructural logics, with explicit translation tools that facilitate cross-logic reasoning and representation.
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 Martín 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.