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Luca Carai

Publications and source records attributed to Luca Carai.

At least 19 recordsLinked to original sources
arXiv2026

A completion of reduced commutative rings

A commutative ring is reduced when it can be embedded into a direct product of fields. While the category of reduced commutative rings plays a fundamental role in affine geometry, it exhibits several structural deficiencies: it admits nonregular monomorphisms and epimorphisms, lacks amalgamation, and is not equationally axiomatizable. In this paper, we simultaneously repair these defects via a canonical completion in which all monomorphisms become regular. This completion is obtained by adjoining weak inverses and weak prime roots, turning the class of reduced commutative rings into a discriminator variety. As a consequence, we obtain an explicit description of dominions in every class of reduced commutative rings containing all fields. This description is strikingly simple compared to that of dominions in the category of all commutative rings, as reflected in the Isbell-Mazet-Silver Zigzag Theorem.

math.RA↗
arXiv2026

A categorical description of simple Beth companions

A pp expansion of a quasivariety $\mathsf{K}$ is said to be simple when it is of the form $\mathsf{K}[\mathscr{L}_\mathcal{F}]$. For instance, when $\mathsf{K}$ has the amalgamation property, all its pp expansions are simple. It is shown that the simple pp expansions of a quasivariety $\mathsf{K}$ coincide with the quasivarieties $\mathsf{M}$ for which the forgetful functor $U \colon \mathsf{M} \to \mathsf{K}$ is well defined and induces an isomorphism from $\mathsf{M}$ to a mono-reflective subcategory of $\mathsf{K}$. As a consequence, if a quasivariety $\mathsf{K}$ possesses a simple Beth companion $\mathsf{M}$, then $\mathsf{M}$ is the unique (up to term equivalence) quasivariety whose monomorphisms are regular that, moreover, satisfy the categorical description of simple pp expansions of $\mathsf{K}$ given above.

math.CT↗
arXiv2026

Bounded depth in Hilbert algebras

Hilbert algebras are the implicative subreducts of Heyting algebras. It is shown that having depth at most n is an equational condition in Hilbert algebras. This generalizes an analogous well-known result in the setting of Heyting algebras.

math.LO↗
arXiv2026

Implicit operations in varieties of commutative monoids

An implicit operation of a class of similar algebras $\mathsf{K}$ is a collection of first order definable partial functions on the members of $\mathsf{K}$ that is globally preserved by homomorphisms. For instance, "taking inverses" can be viewed as a unary implicit operation of the class of all monoids because its graph on a given monoid is defined by the equation $xy \thickapprox 1 \thickapprox yx$ and monoid homomorphisms preserve existing inverses. As this example demonstrates, the implicit operations of a class $\mathsf{K}$ need not be given by a term of $\mathsf{K}$. We show that an equational class of commutative monoids can be expanded with enough implicit operations so that every implicit operation can be interpolated by a family of terms if and only, in each of its members, for every $a$ there exists some $b$ such that $a = a^2b$, i.e., the class consists of inverse monoids. Our methods build on the interaction of the theory of implicit operations with Grillet's description of finitely generated subdirectly irreducible commutative semigroups and the combinatorics deriving from an extension of Isbell's Zigzag Theorem to all equational classes of commutative monoids.

math.RA↗
arXiv2025

Esakia's theorem for the amended monadic intuitionistic calculus

We show that the amended monadic Grzegorczyk logic $\mathsf{M^+Grz}$ is the largest modal companion of the amended monadic intuitionistic logic $\mathsf{M^+IPC}$. Thus, unlike the monadic intuitionistic logic $\mathsf{MIPC}$, Esakia's theorem does extend to $\mathsf{M^+IPC}$.

math.LO↗
arXiv2025

The theory of implicit operations

A family of partial functions of a class of algebras $\mathsf{K}$ is said to be an implicit operation of $\mathsf{K}$ when it is defined by a first order formula and it is preserved by homomorphisms. In this work, we develop the theory of implicit operations from an algebraic standpoint.

math.RA↗
arXiv2025

An addendum to "The theory of implicit operations"

In this addendum to [4], we provide a pair of counterexamples relevant to the theory of implicit operations. More precisely, we exhibit a pp expansion of a variety that fails to be a variety (although it is a quasivariety). Furthermore, we construct a sequence of varieties possessing a congruence preserving Beth companion, but no simple Beth companion.

math.RA↗
arXiv2025

Congruence permutability in quasivarieties

It is shown that a natural notion of congruence permutability for quasivarieties already implies ``being a variety''. The result follows immediately from [3] and the sole aim of this note is to state it explicitly, together with a telegraphic proof.

math.LO↗
arXiv2025

On the lack of colimits in various categories arising in pointfree topology and algebraic logic

We prove that the category of McKinsey-Tarski algebras is not equivalent to a variety of algebras, thus answering a question of Peter Jipsen in the negative. More generally, we show that various categories of BAOs (boolean algebras with an operator), Heyting algebras, and frames with appropriate morphisms between them are not cocomplete. As a consequence, none of these categories is equivalent to a prevariety, let alone a variety.

math.LO↗
arXiv2025

Duality Theory for Bounded Lattices: A Comparative Study

There are numerous generalizations of the celebrated Priestley duality for bounded distributive lattices to the non-distributive setting. The resulting dualities rely on an earlier foundational work of such authors as Nachbin, Birkhoff-Frink, Bruns-Lakser, Hofmann-Mislove-Stralka, and others. We undertake a detailed comparative study of the existing dualities for arbitrary bounded (non-distributive) lattices, including supplying the dual description of bounded lattice homomorphisms where it was lacking. This is achieved by working with relations instead of functions. As a result, we arrive at a landscape of categories that provide various generalizations of the category of Priestley spaces. We provide explicit descriptions of the functors yielding equivalences of these categories, together with explicit dual equivalences with the category of bounded lattices and bounded lattice homomorphisms.

math.LO↗
arXiv2024

Failure of Esakia's theorem in the monadic setting

Esakia's theorem states that Grzegorczyk's logic is the greatest modal companion of intuitionistic propositional calculus. We prove that already the one-variable fragment of intuitionistic predicate calculus does not have a greatest modal companion, yielding that Esakia's theorem fails in the monadic setting.

math.LO↗
arXiv2024

On the universal theory of the free pseudocomplemented distributive lattice

It is shown that the universal theory of the free pseudocomplemented distributive lattice is decidable and a recursive axiomatization is presented. This contrasts with the case of the full elementary theory of the finitely generated free algebras which is known to be undecidable. As a by-product, a description of the finitely generated pseudocomplemented distributive lattices that can be embedded into the free algebra is also obtained.

math.LO↗
arXiv2024

Free algebras and coproducts in varieties of G\"odel algebras

G\"odel algebras are the Heyting algebras satisfying the axiom $(x \to y) \vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe free G\"odel algebras and coproducts of G\"odel algebras. In particular, we realize the Esakia space dual to a G\"odel algebra free over a distributive lattice as the, suitably topologized and ordered, collection of all nonempty closed chains of the Priestley dual of the lattice. This provides a tangible dual description of free G\"odel algebras without any restriction on the number of free generators, which generalizes known results for the finitely generated case. A similar approach allows us to characterize the Esakia spaces dual to coproducts of arbitrary families of G\"odel algebras. We also establish analogous dual descriptions of free algebras and coproducts in every variety of G\"odel algebras. As consequences of these results, we obtain a formula to compute the depth of coproducts of G\"odel algebras and show that all free G\"odel algebras are bi-Heyting algebras.

math.LO↗
arXiv2024

Failure of the Blok-Esakia Theorem in the monadic setting

The Blok-Esakia Theorem establishes that the lattice of superintuitionistic logics is isomorphic to the lattice of extensions of Grzegorczyk's logic. We prove that the Blok-Esakia isomorphism $\sigma$ does not extend to the fragments of the corresponding predicate logics of already one fixed variable. In other words, we prove that $\sigma$ is no longer an isomorphism from the lattice of extensions of the monadic intuitionistic logic to the lattice of extensions of the monadic Grzegorczyk logic.

math.LO↗
arXiv2024

Epimorphisms between finitely generated algebras

A quasivariety has the weak ES property when the epimorphisms between its finitely generated members are surjective. A characterization of quasivarieties with the weak ES property is obtained and a method for detecting failures of this property in quasivarieties with a near unanimity term and in congruence permutable varieties is given. It is also shown that under reasonable assumptions the weak ES property implies arithmeticity. In particular, every filtral variety with the weak ES property is a discriminator variety.

math.LO↗
arXiv2024

A calculus for modal compact Hausdorff spaces

The symmetric strict implication calculus $\mathsf{S^2IC}$ is a modal calculus for compact Hausdorff spaces. This is established through de Vries duality, linking compact Hausdorff spaces with de Vries algebras-complete Boolean algebras equipped with a special relation. Modal compact Hausdorff spaces are compact Hausdorff spaces enriched with a continuous relation. These spaces correspond, via modalized de Vries duality, to upper continuous modal de Vries algebras. In this paper we introduce the modal symmetric strict implication calculus $\mathsf{MS^2IC}$, which extends $\mathsf{S^2IC}$. We prove that $\mathsf{MS^2IC}$ is strongly sound and complete with respect to upper continuous modal de Vries algebras, thereby providing a logical calculus for modal compact Hausdorff spaces. We also develop a relational semantics for $\mathsf{MS^2IC}$ that we employ to show admissibility of various $\Pi_2$-rules in this system.

math.LO↗
arXiv2023

MacNeille completions of subordination algebras

$\mathsf{S5}$-subordination algebras are a natural generalization of de Vries algebras. Recently it was proved that the category $\mathsf{SubS5^S}$ of $\mathsf{S5}$-subordination algebras and compatible subordination relations between them is equivalent to the category of compact Hausdorff spaces and closed relations. We generalize MacNeille completions of boolean algebras to the setting of $\mathsf{S5}$-subordination algebras, and utilize the relational nature of the morphisms in $\mathsf{SubS5^S}$ to prove that the MacNeille completion functor establishes an equivalence between $\mathsf{SubS5^S}$ and its full subcategory consisting of de Vries algebras. We also show that the functor that associates to each $\mathsf{S5}$-subordination algebra the frame of its round ideals establishes a dual equivalence between $\mathsf{SubS5^S}$ and the category of compact regular frames and preframe homomorphisms. Our results are choice-free and provide further insight into Stone-like dualities for compact Hausdorff spaces with various morphisms between them. In particular, we show how they restrict to the wide subcategories of $\mathsf{SubS5^S}$ corresponding to continuous relations and continuous functions between compact Hausdorff spaces.

math.GN↗