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

Abstract

The category 𝔻iffℝ𝔻𝕃′ , whose objects are c-differential residuated distributive lattices that satisfy the condition 𝐂𝐊, is the image of the category ℝ𝔻𝕃, whose objects are residuated distributive lattices, under the categorical equivalence (Kalman functor) 𝐊. The main goal of this paper is to lift this equivalence 𝐊 to the category 𝕄onℝ𝔻𝕃, whose objects are monadic residuated distributive lattices, and the category 𝕄on𝔻iffℝ𝔻𝕃′ , whose objects are pairs formed by an object of 𝔻iffℝ𝔻𝕃′ and a center universal quantifier. Firstly, based on the variety of monadic FLe -algebras, we introduce the concept of monadic residuated lattices and study some of their further algebraic properties, proving that the class of monadic expansions of a residuated distributive lattice 𝐿 and the class of monadic c-differential residuated distributive lattice expansions of 𝐊(𝐿) are in one-to-one correspondence. Subsequently, we further prove a categorical correspondence between the categories 𝕄onℝ𝔻𝕃 and 𝕄on𝔻iffℝ𝔻𝕃′ via the functor 𝐊, which is motivated by an old construction of Kalman. Finally, we provide a finite, nontrivial 2-contextual translation from the equational consequence relation of 𝕄on𝔻iffℝ𝔻𝕃′ into that of 𝕄onℝ𝔻𝕃. The results of this paper not only generalize the work of Sagastume and San Martín in [Mathematical Logic Quarterly, 60(2014), 375–388], but also addresses and overcomes the limitations identified in the works of [Studia Logica, 111(2023), 361–390].