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A categorical equivalence for monadic algebras of first-order substructural logics motivated by Kalman’s construction
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DOI：10.1016/j.fss.2026.110022.png)
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Abstract
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The category , whose objects are c-differential residuated distributive lattices that satisfy the condition CK, is the image of the category , 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 , whose objects are monadic residuated distributive lattices, and the category , whose objects are pairs formed by an object of 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 and 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 into that of . 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].
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