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

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.

• The paper **establishes a categorical equivalence** between the category of monadic residuated distributive lattices and the category **monadic c-differential residuated lattices**, building on Kalman's construction for first-order substructural logics.
• It **generalizes earlier work** on UMV-algebras and corrects technical limitations ..., providing a comprehensive framework that includes new isomorphisms: $\phi(x)=(x,0)$ and $\psi(x) = (x \cup c, \sim x \cup c)$.
• This **equivalency withstands examination against central properties** across these algebraic constructs, ultimately yielding a **commutative cube of categories** highlighting the relations between different algebraic structures and theorem proof techniques.

Background and motivation

The Kalman construction, originally introduced to relate distributive lattices and Kleene algebras, has been developed by Castiglioni, Menni and Sagastume into a categorical equivalence between the category $\mathbb{RDL}$ of residuated distributive lattices and the category $\mathbb{DRDL'}$ of c-differential residuated distributive lattices satisfying the condition $\mathbf{CK}$. Subsequent work lifted this equivalence to monadic MV-algebras (Castiglioni, Lewin and Sagastume) and to UMV-algebras (Sagastume and San Martín). In parallel, Tuyt introduced the variety of monadic FL$_\textrm{e}$-algebras as the algebraic semantics for one-variable fragments of first-order substructural logics over $\mathbf{FL_{ew}$.

The paper under review, by Wang, Wang and Zuluaga Botero (2601.22023), pursues three goals. First, it replaces the "weak monadic residuated distributive lattices" of an earlier paper by Wang et al. — a variety lacking a logical foundation — with monadic residuated lattices in Tuyt's sense, thereby anchoring the construction in the algebraic semantics of monadic fragments of first-order substructural logics. Second, it corrects a technical limitation of that earlier work: the isomorphisms $\alpha(x)=(x,{\sim}x)$ and $\beta(x)=(\lambda(x),\lambda({\sim}x))$ used there are applicable only within involutive residuated lattices, whereas the isomorphisms $\phi(x)=(x,0)$ and $\psi(x)=(x\cup c,{\sim}x\cup c)$ apply generally. Third, it unifies the diagram relating UMV-algebras and UMV$^\bullet$-algebras with the broader residuated-lattice setting, completing a square of functors involving the forgetful functor and the Kalman equivalence.

Monadic residuated lattices

The authors define a monadic residuated lattice as a triple $\mathbb{DRDL'}$0 where $\mathbb{DRDL'}$1 is a residuated lattice and $\mathbb{DRDL'}$2, $\mathbb{DRDL'}$3 satisfy identities (M1)–(M3) and (M6)–(M7), mirroring Halmos's monadic Boolean algebras but without requiring both quantifiers as primitive where they are interdefinable. A useful simplification is established early: conditions (M4) ($\mathbb{DRDL'}$4) and (M5) ($\mathbb{DRDL'}$5) follow from the remaining axioms, so the axiomatization can be reduced. The class $\mathbb{DRDL'}$6 forms a variety, and an extensive list of derived identities is proved, including the Galois connection $\mathbb{DRDL'}$7 iff $\mathbb{DRDL'}$8 and the interaction laws $\mathbb{DRDL'}$9 and $\mathbf{CK}$0.

Two term-equivalence results connect this variety to known structures:

• The subvariety of $\mathbf{CK}$1 satisfying involution $\mathbf{CK}$2 is term-equivalent to the variety $\mathbf{CK}$3 of monadic involutive residuated lattices, where only $\mathbf{CK}$4 is primitive and $\mathbf{CK}$5.
• The subvariety determined by (INV), divisibility (DIV) and prelinearity (PRE) is term-equivalent to the variety $\mathbf{CK}$6 of UMV-algebras (MV-algebras with U-operators).

These results have a direct consequence for the categorical program: since UMV-algebras embed into monadic residuated lattices via this term equivalence, the Sagastume–San Martín lifting from $\mathbf{CK}$7 becomes a special case of the more general construction developed here.

Monadic c-differential residuated lattices

Given a monadic residuated distributive lattice $\mathbf{CK}$8, the authors define an operator on the Kalman structure $\mathbf{CK}$9 by

$_\textrm{e}$0

A substantial theorem verifies sixteen properties of $_\textrm{e}$1: well-definedness (using $_\textrm{e}$2), preservation of bounds and the center $_\textrm{e}$3, meet-compatibility, the adjunction-type identity $_\textrm{e}$4, monotonicity, idempotence, and compatibility with the center operations $_\textrm{e}$5, $_\textrm{e}$6, $_\textrm{e}$7. These motivate the central new definition: a center universal quantifier on a c-differential residuated lattice is a unary operator $_\textrm{e}$8 satisfying seven identities, including $_\textrm{e}$9 and the three center-compatibility clauses. The resulting algebras, monadic c-differential residuated lattices, again form a variety.

A remark worth noting is that center universal quantifiers are strictly more restrictive than ordinary universal quantifiers on involutive residuated lattices: the operator $\mathbf{FL_{ew}$0 satisfies the latter axioms but fails $\mathbf{FL_{ew}$1 whenever $\mathbf{FL_{ew}$2. This distinction justifies introducing a genuinely new notion rather than importing the involutive case wholesale.

The key correspondence result states that the assignment $\mathbf{FL_{ew}$3 is a bijection, up to isomorphism, between monadic expansions of a residuated distributive lattice $\mathbf{FL_{ew}$4 and monadic c-differential expansions of $\mathbf{FL_{ew}$5. The converse direction recovers $\mathbf{FL_{ew}$6 on $\mathbf{FL_{ew}$7 as $\mathbf{FL_{ew}$8 and defines $\mathbf{FL_{ew}$9; uniqueness follows because existential quantifiers are uniquely determined by universal ones via the Galois connection. An implication of this result is that no information about the quantifier structure is lost or gained in passing through the Kalman functor — the monadic structure transfers exactly.

The categorical equivalence

The main theorem extends the functors $\alpha(x)=(x,{\sim}x)$0 to the monadic setting. On morphisms, $\alpha(x)=(x,{\sim}x)$1 preserves the quantifier structure, and $\alpha(x)=(x,{\sim}x)$2 restricts homomorphisms of monadic c-differential residuated distributive lattices to their upper cones $\alpha(x)=(x,{\sim}x)$3, which carry a canonical monadic residuated distributive lattice structure with $\alpha(x)=(x,{\sim}x)$4.

The proof strategy uses precisely the isomorphisms $\alpha(x)=(x,{\sim}x)$5 and $\alpha(x)=(x,{\sim}x)$6 that the authors identify as the correct general-purpose tools. It is shown that $\alpha(x)=(x,{\sim}x)$7 is an isomorphism in $\alpha(x)=(x,{\sim}x)$8 (preserving both $\alpha(x)=(x,{\sim}x)$9 and $\beta(x)=(\lambda(x),\lambda({\sim}x))$0), that $\beta(x)=(\lambda(x),\lambda({\sim}x))$1 preserves the center universal quantifier, and that $\beta(x)=(\lambda(x),\lambda({\sim}x))$2 is surjective exactly when the condition $\beta(x)=(\lambda(x),\lambda({\sim}x))$3 holds. Consequently:

Main theorem. The functors $\beta(x)=(\lambda(x),\lambda({\sim}x))$4 and $\beta(x)=(\lambda(x),\lambda({\sim}x))$5 establish a categorical equivalence between $\beta(x)=(\lambda(x),\lambda({\sim}x))$6 and $\beta(x)=(\lambda(x),\lambda({\sim}x))$7, with natural isomorphisms $\beta(x)=(\lambda(x),\lambda({\sim}x))$8 and $\beta(x)=(\lambda(x),\lambda({\sim}x))$9.

This simultaneously generalizes the Sagastume–San Martín result for UMV-algebras (which cannot simplify their source category, since both sides there are involutive) and repairs the limitation of the earlier weak-monadic construction. Combined with Corollary 3.8, the categories $\phi(x)=(x,0)$0 and $\phi(x)=(x,0)$1 sit inside $\phi(x)=(x,0)$2 and $\phi(x)=(x,0)$3 respectively, so the two Kalman liftings — the MV-level one and the residuated-lattice-level one — are related by inclusion functors in a commutative cube of categories, completing the program sketched in the introduction.

A 2-contextual translation

As an application, the paper exploits Moraschini's correspondence between nontrivial adjunctions and $\phi(x)=(x,0)$4-contextual translations of equational consequence relations. The technical core is an explicit description of free algebras: $\phi(x)=(x,0)$5 is isomorphic to the quotient of the free $\phi(x)=(x,0)$6-algebra on $\phi(x)=(x,0)$7 generators $\phi(x)=(x,0)$8 by the congruence forcing $\phi(x)=(x,0)$9 for each $\psi(x)=(x\cup c,{\sim}x\cup c)$0. The proof proceeds by induction using preservation of coproducts by both $\psi(x)=(x\cup c,{\sim}x\cup c)$1 and the free-algebra functor.

From this description the authors extract an explicit 2-translation $\psi(x)=(x\cup c,{\sim}x\cup c)$2, defined componentwise on each operation symbol; notably,

$\psi(x)=(x\cup c,{\sim}x\cup c)$3

with the clause verified via the computation $\psi(x)=(x\cup c,{\sim}x\cup c)$4. Taking the context $\psi(x)=(x\cup c,{\sim}x\cup c)$5, the pair $\psi(x)=(x\cup c,{\sim}x\cup c)$6 yields a finite, nontrivial 2-contextual translation from $\psi(x)=(x\cup c,{\sim}x\cup c)$7 to $\psi(x)=(x\cup c,{\sim}x\cup c)$8, and faithfulness of $\psi(x)=(x\cup c,{\sim}x\cup c)$9 gives the soundness-and-completeness statement: equational consequence in $^\bullet$0 holds exactly when its $^\bullet$1-image, together with the doubled-variable constraints $^\bullet$2, holds in $^\bullet$3. In other words, equational reasoning about monadic c-differential residuated distributive lattices can be faithfully simulated inside plain monadic residuated distributive lattices at the cost of duplicating variables.

Limitations and open questions

The equivalence is established specifically for distributive residuated lattices satisfying integrality, and the condition $^\bullet$4 is essential — without it, $^\bullet$5 fails to be surjective and only an adjunction, not an equivalence, is available. The authors state explicitly that the natural next step, an equivalence between monadic distributive FL$^\bullet$6-algebras and monadic distributive c-differential FL$^\bullet$7-algebras, remains open: the Kalman functor $^\bullet$8 is not suitable for these categories in general, and reconstructing the underlying non-monadic equivalence between $^\bullet$9 and $\mathbb{DRDL'}$00 is identified as the critical obstacle. Whether the contextual translation machinery extends to that setting is likewise unresolved.

Conclusion

The paper lifts the Kalman categorical equivalence from residuated distributive lattices to their monadic counterparts, replacing a logically unmotivated "weak monadic" variety with Tuyt's monadic residuated lattices and correcting a generality defect in the earlier proof technique. The resulting equivalence between $\mathbb{DRDL'}$01 and $\mathbb{DRDL'}$02 both subsumes the UMV-algebra case and situates it within a commutative diagram of categories, and it yields, via Moraschini's framework, an explicit finite 2-contextual translation showing that the equational theory of monadic c-differential residuated distributive lattices is faithfully interpretable in that of monadic residuated distributive lattices.