arXiv subjects

Miguel Campercholi

Publications and source records attributed to Miguel Campercholi.

9 recordsLinked to original sources

arXiv2026

Global representations of algebras with a near-unanimity term

Global representations are subdirect representations satisfying a sheaf-like local-to-global patching principle. We develop a unified and simplified theory of such representations. For quasivarieties with a near-unanimity term, this framework recovers the main classical representation theorems through a common argument and yields a general representation by factors of bounded subdirect width. When the relatively subdirectly irreducible members form a universal class, we obtain an optimal result: every relatively congruence-distributive algebra admits a global representation by relatively globally indecomposable factors, which we characterize explicitly. We also provide a converse: global representations by factors of bounded relative subdirect width force the existence of a near-unanimity term. Adding semisimplicity to the hypotheses of the main theorem yields optimal representation results for filtral quasivarieties and dual discriminator varieties, together with a simple description of the globally indecomposable algebras. The technical engine behind these results is a new infinitary extension of the congruence-system component of the Baker--Pixley theorem.

math.RA↗

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.

cs.CC↗

arXiv2023

Two algorithms to decide Quantifier-free Definability in Finite Algebraic Structures

This work deals with the definability problem by quantifier-free first-order formulas over a finite algebraic structure. We show the problem to be coNP-complete and present two decision algorithms based on a semantical characterization of definable relations as those preserved by isomorphisms of substructures, the second one also providing a formula in the positive case. Our approach also includes the design of an algorithm that computes the isomorphism type of a tuple in a finite algebraic structure. Proofs of soundness and completeness of the algorithms are presented, as well as empirical tests assessing their performances.

cs.LO↗

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.

math.LO↗

arXiv2019

The Complexity of Definability by Open First-Order Formulas

In this article we formally define and investigate the computational complexity of the Definability Problem for open first-order formulas (i.e., quantifier free first-order formulas) with equality. Given a logic $\mathbf{\mathcal{L}}$, the $\mathbf{\mathcal{L}}$-Definability Problem for finite structures takes as input a finite structure $\mathbf{A}$ and a target relation $T$ over the domain of $\mathbf{A}$, and determines whether there is a formula of $\mathbf{\mathcal{L}}$ whose interpretation in $\mathbf{A}$ coincides with $T$. We show that the complexity of this problem for open first-order formulas (open definability, for short) is coNP-complete. We also investigate the parametric complexity of the problem, and prove that if the size and the arity of the target relation $T$ are taken as parameters then open definability is $\mathrm{coW}[1]$-complete for every vocabulary $τ$ with at least one, at least binary, relation.

cs.CC↗

arXiv2017

The Lattice of Congruences of a Finite Line Frame

Let $\mathbf{F}=\left\langle F,R\right\rangle $ be a finite Kripke frame. A congruence of $\mathbf{F}$ is a bisimulation of $\mathbf{F}$ that is also an equivalence relation on F. The set of all congruences of $\mathbf{F}$ is a lattice under the inclusion ordering. In this article we investigate this lattice in the case that $\mathbf{F}$ is a finite line frame. We give concrete descriptions of the join and meet of two congruences with a nontrivial upper bound. Through these descriptions we show that for every nontrivial congruence $ρ$, the interval $[\mathrm{Id_{F},ρ]}$ embeds into the lattice of divisors of a suitable positive integer. We also prove that any two congruences with a nontrivial upper bound permute.

math.LO↗

arXiv2016

Epic substructures and primitive positive functions

For $\mathbf{A}\leq\mathbf{B}$ first order structures in a class $\mathcal{K}$, say that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in $\mathcal{K}$ if for every $\mathbf{C}\in\mathcal{K}$ and all homomorphisms $g,g^{\prime}:\mathbf{B}\rightarrow\mathbf{C}$, if $g$ and $g'$ agree on $A$, then $g=g'$. We prove that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in a class $\mathcal{K}$ closed under ultraproducts if and only if $A$ generates $\mathbf{B}$ via operations definable in $\mathcal{K}$ with primitive positive formulas. Applying this result we show that a quasivariety of algebras $\mathcal{Q}$ with an $n$-ary near-unanimity term has surjective epimorphisms if and only if $\mathbb{SP}_{n}\mathbb{P}_{u}(\mathcal{Q}_{RSI})$ has surjective epimorphisms. It follows that if $\mathcal{F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by $\mathcal{F}$ has surjective epimorphisms.

math.LO↗

arXiv2015

Semantical conditions for the definability of functions and relations

Let $\mathcal{L}\subseteq \mathcal{L}^{\prime }$ be first order languages, let $R\in \mathcal{L}^{\prime }-\mathcal{L}$ be a relation symbol, and let $% \mathcal{K}$ be a class of $\mathcal{L}^{\prime }$-structures. In this paper we present semantical conditions equivalent to the existence of an $\mathcal{L}$-formula $φ\left( \vec{x}\right) $ such that $\mathcal{K}\vDash φ(\vec{x})\leftrightarrow R(\vec{x})$, and $φ$ has a specific syntactical form (e.g., quantifier free, positive and quantifier free, existential horn, etc.). For each of these definability results for relations we also present an analogous version for the definability of functions. Several applications to natural definability questions in universal algebra have been included; most notably definability of principal congruences. The paper concludes with a look at term-interpolation in classes of structures with the same techniques used for definability. Here we obtain generalizations of two classical term-interpolation results: Pixley's theorem for quasiprimal algebras, and the Baker-Pixley Theorem for finite algebras with a majority term.

math.LO↗

arXiv2014

On structural completeness vs almost structural completeness problem: A discriminator varieties case study

We study the following problem: Determine which almost structurally complete quasivarieties are structurally complete. We propose a general solution to this problem and then a solution in the semisimple case. As a consequence, we obtain a characterization of structurally complete discriminator varieties. An interesting corollary in logic follows: Let $L$ be a consistent propositional logic/deductive system in the language with formulas for verum, which is a theorem, and falsum, which is not a theorem. Assume also that $L$ has an adequate semantics given by a discriminator variety. Then $L$ is structurally complete if and only if it is maximal. All such logics/deductive systems are almost structurally complete.

math.LO↗