arXiv subjects

Carles Cardó

Publications and source records attributed to Carles Cardó.

11 recordsLinked to original sources

arXiv2026

On semigroups and groupoids with minimal probabilistic spectrum

The equational probabilistic spectrum of a finite algebra is the set of probabilities with which equations are satisfied in the algebra. We study algebras with minimal spectrum, that is, spectra consisting only of the values $1$ and $1/|\A|$. We show that, apart from trivial cases, groupoids with minimal spectrum are quasigroups. We further prove that several weak associativity conditions collapse into full associativity, and hence into group structure. Finally, we obtain a complete classification of semigroups with minimal spectrum.

math.LO↗

arXiv2026

Finite versus uncountable convex lattices from point configurations

We study the smallest convex lattice generated by a finite set of points. To analyze this structure, we introduce the notion of a point configuration, defined via the relative lattice. Under a suitable completeness condition, this lattice becomes a combinatorial counterpart of the convex lattice and is therefore easier to handle. We investigate the enumeration of these structures and prove that, while the number of relative lattices is always finite, the number of convex lattices is uncountable for $n \geq 6$.

math.CO↗

arXiv2026

Probabilistic equational spectrum, primality and approximation in finite algebras

We define the probability of an equation in a finite algebra as the proportion of tuples in its domain that satisfy it. We call the probabilistic spectrum of an algebra the set of probability values obtained when the equation varies. We study fundamental properties of this spectrum, such as density and limit points, and show that its structure is related to several notions of primality of an algebra. We introduce a quantitative measure of primality $\Prim(\A)\in[0,1]$ that characterizes the functional approximation capacity. We show that the degree of primality is related to the size of the spectrum. We also prove that all non-primal two-element algebras satisfy the universal bound $\Prim(\A)\le 1/2$.

math.LO↗

arXiv2024

Arithmetic and $k$-maximality of the cyclic free magma

We survey free magmas and we explore the structure of their submagmas. By equipping the cyclic free magma with a second distributive operation we obtain a ringoid-like structure with some primitive arithmetical properties. A submagma is $k$-maximal when there are only $k-1$ submagmas between it and the free magma itself. These two tools, arithmetic and maximality, allow us to study the lattice of the submagmas of a free magma.

math.RA↗

arXiv2024

Equidecomposable magmas

A magma is called equidecomposable when the operation is injective, or, in other words, if $x+y=x'+y'$ implies that $x=x'$ and $y=y'$. A magma is free iff it is equidecomposable and graded, hence the notion of equidecomposability is very related to the notion of freeness although it is not sufficient. We study main properties of such magmas. In particular, an alternative characterization of freeness, which uses a weaker condition, is proved. We show how equidecomposable magmas can be split into two disjoint submagmas, one of which is free. Certain tranformations on finite presentations permit to obtain a reduced form which allows us identify all the finite presented equidecomposable magmas up to isomorphisms.

math.RA↗

arXiv2024

Convex hull lattices point generated

The simplest way to generate a lattice of convex sets is to consider an initial set of points and draw segments, triangles, and any convex hull from it, then intersect them to obtain new points, and so forth. The result is an infinite lattice for most sets, while only a few initial sets of points perform a finite lattice. By giving an adequate notion of the configuration of points, we identify which sets in the plane define a finite convex hull lattice: four regular families and one sporadic configuration. We explore configurations in the space and higher dimensions.

math.CO↗

arXiv2024

A look at the Kolmogorov complexity of finite groupoids and algebras

The incompressibility method is a counting argument in the framework of algorithmic complexity that permits discovering properties that are satisfied by most objects of a class. This paper gives a preliminary insight into Kolmogorov's complexity of groupoids and some algebras. The incompressibility method shows that almost all the groupoids are asymmetric and simple: Only trivial or constant homomorphisms are possible. However, highly random groupoids allow subgroupoids with interesting restrictions that reveal intrinsic structural properties. We also study the issue of the algebraic varieties and wonder which equational identities allow randomness.

cs.IT↗

arXiv2024

Growth and density in free groupoids

The density of a subgroupoid with respect to a free groupoid is defined as the asymptotic ratio of their growths. This notion can be interpreted as a generalisation of the index's inverse for groups or as the probability of an element belonging to a subgroupoid. This more combinatorial strategy shows a richer picture of free groupoids than the bare algebraic perspective. We study the growth and density of several subgroupoids of a free groupoid. In addition, some aspects of enumeration related to the Motzkin paths are shown.

math.CO↗

arXiv2024

A note on the probability of a groupoid having deficient sets

A subset $X$ of a groupoid is said to be deficient if $|X \cdot X|\leq |X|$. It is well-known that the probability that a random groupoid has a deficient $t$-element set with $t\geq 3$ is zero. However, as conjectured in [4], we show that the probability is not zero in the case of sets of two elements and calculate the exact value. We explore some generalisations on deficient sets and their likelihoods.

math.GR↗

arXiv2024

Anti-Context-Free languages

Context-free languages can be characterized in several ways. This article studies projective linearisations of languages of simple dependency trees, i.e., dependency trees in which a node can govern at most one node with a given syntactic function. We prove that the projective linearisations of local languages of simple dependency trees coincide with the context-free languages. Simple dependency trees suggest alternative dual notions of locality and projectivity, which permits defining a dual language for each context-free language. We call this new class of languages anti-context-free. These languages are related to some linguistic constructions exhibiting the so-called cross-serial dependencies that were historically important for the development of computational linguistics. We propose that this duality could be a relevant linguistic phenomenon.

cs.FL↗

arXiv2023

On semi-Peano algebras

A semi-Peano algebra is an algebra for which each operation is injective, and the images of the operations are pairwise disjoint. The most straightforward non-trivial kind of finitely presented semi-Peano algebra are algebras with a single unary operation. There are two possible directions of generalization: algebras with a single operation of any arity, and algebras with several unary operations. The former can be solved easily by adapting results on equidecomposable groupoids from [2]. However, the second way is somewhat different. We will show that a finitely presented multi-unary semi-Peano algebra is the free product of cyclic semi-Peano algebras and that a unique relation defines such cyclic algebras. In addition, we will characterize each cyclic algebra up to isomorphism.

math.RA↗