arXiv subjects

Diego Castaño

Publications and source records attributed to Diego Castaño.

10 recordsLinked to original sources

arXiv2026

Conuclei on varieties of hoops

A conucleus $δ$ on a partially ordered monoid $\mathbf{A}$ is an interior operator that satisfies $δ(a) \cdot δ(b) \leq δ(a \cdot b)$ and $δ(a) \cdot δ(1) = δ(a)$ for all $a,b \in A$. A conucleus is multiplicative if the equality $δ(a \cdot b) = δ(a) \cdot δ(b)$ holds for all $a,b \in A$. In this article we focus on the study of conuclei on hoops, structures which generalize well-known classes of algebras, such as the class of MV-algebras and BL-algebras. Among several results, we provide a Glivenko-type theorem for conuclei. Special emphasis is given to term definable conuclei. The main result of this article is an explicit description of all terms that define a multiplicative conucleus on every structure of an arbitrary proper variety of Wajsberg hoops. We also show that the problem of finding terms that define (multiplicative) conuclei on a variety of basic hoops or BL-algebras is equivalent to finding such terms on some variety or some pair of varieties of Wajsberg hoops. We provide nontrivial interesting examples.

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arXiv2024

Terms that define nuclei on residuated lattices: a case study of BL-algebras

A nucleus $γ$ on a (bounded commutative integral) residuated lattice $\mathbf{A}$ is a closure operator that satisfies the inequality $γ(a) \cdot γ(b) \leq γ(a \cdot b)$ for all $a,b \in A$. In this article, among several results, a description of an arbitrary nucleus on a residuated lattice is given. Special attention is given to terms that define a nucleus on every structure of a variety, as a means of generalizing the double negation operation. Some general results about these terms are presented, together with examples. The main result of this article consists of the description of all terms of this kind for every given subvariety of BL-algebras. We exhibit interesting nontrivial examples.

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arXiv2024

The algebraic semantics for the one-variable monadic fragment of the predicate logic $\mathcal{G}\forall_{\sim}$

In this article we characterize the equivalent algebraic semantics for the one-variable monadic fragment of the first-order logic ${\cal G} \forall_{\sim}$ defined by F. Esteva, L. Godo, P. Hájek and M. Navara in Residuated fuzzy logics with an involutive negation, Archive for Mathematical Logic 39 (2000). To this end, we first introduce the variety $\mathbb{MG}_{\sim}$ as a certain class of Gödel algebras endowed with two monadic operators and a De Morgan negation. We study its basic properties, determine its subdirectly irreducible members and prove that this variety has the finite embeddabilty property. In particular, we prove that a special subvariety $\mathbb{CMG}_{\sim}$ of $\mathbb{MG}_\sim$ is exactly the desired equivalent algebraic semantics; this is done via a functional representation of finite subdirectly irreducible algebras.

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arXiv2024

Strong standard completeness theorems for S5-modal Lukasiewicz logics

We study the S5-modal expansion of the logic based on the Lukasiewicz t-norm. We exhibit a finitary propositional calculus and show that it is finitely strongly complete with respect to this logic. This propositional calculus is then expanded with an infinitary rule to achieve strong completeness. These results are derived from properties of monadic MValgebras: functional representations of simple and finitely subdirectly irreducible algebras, as well as the finite embeddability property. We also show similar completeness theorems for the extension of the logic based on models with bounded universe.

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arXiv2024

Strong completeness for the predicate logic of the continuous t-norms

The axiomatic system introduced by Hájek axiomatizes first-order logic based on BL-chains. In this study, we extend this system with the axiom $(\forall x ϕ)^2 \leftrightarrow \forall x ϕ^2$ and the infinitary rule \[ \frac{ϕ\vee (α\to β^n):n \in \mathbb{N}}{ϕ\vee (α\to α\& β)} \] to achieve strong completeness with respect to continuous t-norms.

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arXiv2023

The complexity of the Chinese Remainder Theorem

The Chinese Remainder Theorem for the integers says that every system of congruence equations is solvable as long as the system satisfies an obvious necessary condition. This statement can be generalized in a natural way to arbitrary algebraic structures using the language of Universal Algebra. In this context, an algebra is a structure of a first-order language with no relation symbols, and a congruence on an algebra is an equivalence relation on its base set compatible with its fundamental operations. A tuple of congruences of an algebra is called a Chinese Remainder tuple if every system involving them is solvable. In this article we study the complexity of deciding whether a tuple of congruences of a finite algebra is a Chinese Remainder tuple. This problem, which we denote CRT, is easily seen to lie in coNP. We prove that it is actually coNP-complete and also show that it is tractable when restricted to several well-known classes of algebras, such as vector spaces and distributive lattices. The polynomial algorithms we exhibit are made possible by purely algebraic characterizations of Chinese Remainder tuples for algebras in these classes, which constitute interesting results in their own right. Among these, an elegant characterization of Chinese Remainder tuples of finite distributive lattices stands out. Finally, we address the restriction of CRT to an arbitrary equational class $\mathcal{V}$ generated by a two-element algebra. Here we establish an (almost) dichotomy by showing that, unless $\mathcal{V}$ is the class of semilattices, the problem is either coNP-complete or tractable.

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arXiv2020

Algebraic expansions of logics and algebras and a case study of Abelian l-groups and perfect MV-algebras

An algebraically expandable (AE) class is a class of algebraic structures axiomatizable by sentences of the form $\forall \exists! \land p = q$. For a logic $L$ algebraized by a quasivariety $\mathcal{Q}$ we show that the AE-subclasses of $\mathcal{Q}$ correspond to certain natural expansions of $L$, which we call {\em algebraic expansions}. These turn out to be a special case of the expansions by implicit connectives studied by X. Caicedo. We proceed to characterize all the AE-subclasses of Abelian $\ell$-groups and perfect MV-algebras, thus fully describing the algebraic expansions of their associated logics.

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arXiv2020

Subvarieties of Pseudocomplemented Kleene Algebras

In this paper we study the subdirectly irreducible algebras in the variety ${\cal PCDM}$ of pseudocomplemented De Morgan algebras by means of their De Morgan $p$-spaces. We introduce the notion of $body$ of an algebra ${\bf L} \in {\cal PCDM}$ and determine $Body({\bf L})$ when ${\bf L}$ is subdirectly irreducible. As a consequence of this, in the case of pseudocomplemented Kleene algebras, three special subvarieties arise naturally, for which we give explicit identities that characterize them. We also introduce a subvariety ${\cal BPK}$ of ${\cal PCDM}$, namely the variety of $bundle$ $pseudocomplemented$ $Kleene$ $algebras$, determine the whole subvariety lattice and find explicit equational bases for each of the subvarieties. In addition, we study the subvariety ${\cal BPK}_0$ of ${\cal BPK}$ generated by the simple members of ${\cal BPK}$, determine the structure of the free algebra over a finite set and their finite weakly projective algebras.

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arXiv2020

An algebraic study of S5-modal Gödel logic

In this paper we continue the study of the variety $\mathbb{MG}$ of monadic Gödel algebras. These algebras are the equivalent algebraic semantics of the S5-modal expansion of Gödel logic, which is equivalent to the one-variable monadic fragment of first-order Gödel logic. We show three families of locally finite subvarieties of $\mathbb{MG}$ and give their equational bases. We also introduce a topological duality for monadic Gödel algebras and, as an application of this representation theorem, we characterize congruences and give characterizations of the locally finite subvarieties mentioned above by means of their dual spaces. Finally, we study some further properties of the subvariety generated by monadic Gödel chains: we present a characteristic chain for this variety, we prove that a Glivenko-type theorem holds for these algebras and we characterize free algebras over $n$ generators.

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arXiv2016

Monadic BL-algebras: the equivalent algebraic semantics of Hájek's monadic fuzzy logic

In this article we introduce the variety of monadic BL-algebras as BL-algebras endowed with two monadic operators $\forall$ and $\exists$. After a study of the basic properties of this variety we show that this class is the equivalent algebraic semantics of the monadic fragment of Hájek's basic predicate logic. In addition, we start a systematic study of the main subvarieties of monadic BL-algebras, some of which constitute the algebraic semantics of well-known monadic logics: monadic Gödel logic and monadic Łukasiewicz logic. In the last section we give a complete characterization of totally ordered monadic BL-algebras.

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