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Erkko Lehtonen

Publications and source records attributed to Erkko Lehtonen.

At least 19 recordsLinked to original sources
arXiv2025

Clonoids of Boolean functions with a linear source clone and a semilattice or 0- or 1-separating target clone

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of Boolean functions. Namely, when $C_1$ is a subclone (a proper subclone, resp.) of the clone of all linear (affine) functions and $C_2$ is a subclone of the clone generated by a semilattice operation and constants (a subclone of the clone of all $0$- or $1$-separating functions, resp.), then the lattice of $(C_1,C_2)$-clonoids is uncountable. Combining this fact with several earlier results, we obtain a complete classification of the cardinalities of the lattices of $(C_1,C_2)$-clonoids for all pairs $(C_1,C_2)$ of clones on $\{0,1\}$.

math.CO↗
arXiv2024

Clonoids of Boolean functions with essentially unary, linear, semilattice, or 0- or 1-separating source and target clones

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions for certain pairs $(C_1,C_2)$ of clones of essentially unary, linear, or $0$- or $1$-separating functions or semilattice operations. When such a $(C_1,C_2)$-clonoid lattice is uncountable, the proof is in most cases based on exhibiting a countably infinite family of functions with the property that distinct subsets thereof always generate distinct $(C_1,C_2)$-clonoids. In the cases when the lattice is finite, we enumerate the corresponding $(C_1,C_2)$-clonoids. We also provide a summary of the known results on cardinalities of $(C_1,C_2)$-clonoid lattices of Boolean functions.

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arXiv2024

Near-unanimity-closed minions of Boolean functions

The near-unanimity-closed minions of Boolean functions, i.e., the clonoids whose target algebra contains a near-unanimity function, are completely described. The key concept towards this result is the minorant-minor partial order and its order ideals.

math.RA↗
arXiv2024

Clonoids of Boolean functions with a monotone or discriminator source clone

Extending Sparks's theorem, we determine the cardinality of the lattice of $(C_1,C_2)$-clonoids of Boolean functions in the cases where the target clone $C_2$ is the clone of projections. Moreover, we explicitly describe the $(C_1,C_2)$-clonoids of Boolean functions in the cases where the source clone $C_1$ is one of the four clones of monotone functions or contains the discriminator function.

math.CO↗
arXiv2024

Associative-commutative spectra for some varieties of groupoids

The associative spectrum of a groupoid (i.e., a set with a binary operation) measures its nonassociativity while the associative-commutative spectrum measures both nonassociativity and noncommutativity of the groupoid. The two spectra are also the coefficients of the Hilbert series of certain operads. We establish upper bounds for the two spectra of various varieties of groupoids defined by different sets of identities and provide examples (often groupoids with three elements) for which the upper bounds are achieved. Our results have connections to many interesting combinatorial objects and integer sequences and naturally lead to some questions for future studies.

math.CO↗
arXiv2023

Associativity conditions for linear quasigroups and equivalence relations on binary trees

We characterise the bracketing identities satisfied by linear quasigroups with the help of certain equivalence relations on binary trees that are based on the left and right depths of the leaves modulo some integers. The numbers of equivalence classes of $n$-leaf binary trees are variants of the Catalan numbers, and they form the associative spectrum (a kind of measure of non-associativity) of a quasigroup.

math.CO↗
arXiv2023

$S$-preclones and the Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$, Part I

We consider $S$-operations $f \colon A^{n} \to A$ in which each argument is assigned a signum $s \in S$ representing a "property" such as being order-preserving or order-reversing with respect to a fixed partial order on $A$. The set $S$ of such properties is assumed to have a monoid structure reflecting the behaviour of these properties under the composition of $S$-operations (e.g., order-reversing composed with order-reversing is order-preserving). The collection of all $S$-operations with prescribed properties for their signed arguments is not a clone (since it is not closed under arbitrary identification of arguments), but it is a preclone with special properties, which leads to the notion of $S$-preclone. We introduce $S$-relations $\varrho = (\varrho_{s})_{s \in S}$, $S$-relational clones, and a preservation property ($f \mathrel{\stackrel{S}{\triangleright}} \varrho$), and we consider the induced Galois connection ${}^S\mathrm{Pol}$-${}^S\mathrm{Inv}$. The $S$-preclones and $S$-relational clones turn out to be exactly the closed sets of this Galois connection. We also establish some basic facts about the structure of the lattice of all $S$-preclones on $A$.

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arXiv2023

The associative-commutative spectrum of a binary operation

We initiate the study of a quantitative measure for the failure of a binary operation to be commutative and associative. We call this measure the associative-commutative spectrum as it extends the so-called associative spectrum (also known as the subassociativity type), which measures the nonassociativity of a binary operation. In fact, the associative-commutative spectrum (resp. associative spectrum) is the cardinality of the symmetric (resp. nonsymmetric) operad obtained naturally from a groupoid (a set with a binary operation). In this paper we provide some general results on the associative-commutative spectrum, precisely determine this measure for certain binary operations, and propose some problems for future study.

math.CO↗
arXiv2022

Stability of Boolean function classes with respect to clones of linear functions

We consider classes of Boolean functions stable under compositions both from the right and from the left with clones. Motivated by the question how many properties of Boolean functions can be defined by means of linear equations, we focus on stability under compositions with the clone of linear idempotent functions. It follows from a result by Sparks that there are countably many such linearly definable classes of Boolean functions. In this paper, we refine this result by completely describing these classes. This work is tightly related with the theory of function minors, stable classes, clonoids, and hereditary classes, topics that have been widely investigated in recent years by several authors including Maurice Pouzet and his coauthors.

math.RA↗
arXiv2022

Galois theory for analogical classifiers

Analogical proportions are 4-ary relations that read "A is to B as C is to D". Recent works have highlighted the fact that such relations can support a specific form of inference, called analogical inference. This inference mechanism was empirically proved to be efficient in several reasoning and classification tasks. In the latter case, it relies on the notion of analogy preservation. In this paper, we explore this relation between formal models of analogy and the corresponding classes of analogy preserving functions, and we establish a Galois theory of analogical classifiers. We illustrate the usefulness of this Galois framework over Boolean domains, and we explicitly determine the closed sets of analogical classifiers, i.e., classifiers that are compatible with the analogical inference, for each pair of Boolean analogies.

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arXiv2021

HS-stability and complex products in involution semigroups

When does the complex product of a given number of subsets of a group generate the same subgroup as their union? We answer this question in a more general form by introducing HS-stability and characterising the HS-stable involution subsemigroup generated by a subset of a given involution semigroup. We study HS-stability for the special cases of regular ${}^{*}$-semigroups and commutative involution semigroups.

math.RA↗
arXiv2021

Majority-closed minions of Boolean functions

The 93 minions of Boolean functions stable under left composition with the clone of self-dual monotone functions are described. As an easy consequence, all $(C_1,C_2)$-stable classes of Boolean functions are determined for an arbitrary clone $C_1$ and for any clone $C_2$ containing the clone of self-dual monotone functions.

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arXiv2021

Reconstructing Young Tableaux

This paper completely characterizes the standard Young tableaux that can be reconstructed from their sets or multisets of $1$-minors. In particular, any standard Young tableau with at least $5$ entries can be reconstructed from its set of $1$-minors.

math.CO↗
arXiv2020

Associative spectra of graph algebras I. Foundations, undirected graphs, antiassociative graphs

Associative spectra of graph algebras are examined with the help of homomorphisms of DFS trees. Undirected graphs are classified according to the associative spectra of their graph algebras; there are only three distinct possibilities: constant 1, powers of 2, and Catalan numbers. Associative and antiassociative digraphs are described, and associative spectra are determined for certain families of digraphs, such as paths, cycles, and graphs on two vertices.

math.CO↗
arXiv2020

Reflections and powers of multisorted minions

Classes of multisorted minions closed under extensions, reflections, and direct powers are considered from a relational point of view. As a generalization of a result of Barto, Opršal, and Pinsker, the closure of a multisorted minion is characterized in terms of constructions on multisorted relation pairs which are invariant for minions.

math.RA↗
arXiv2019

On associative operations on commutative integral domains

We describe the associative multilinear polynomial functions over commutative integral domains. This extends Marichal and Mathonet's result on infinite integral domains and provides a new proof of Andres's classification of two-element $n$-semigroups.

math.RA↗
arXiv2019

Graph quasivarieties

Introduced by C. R. Shallon in 1979, graph algebras establish a useful connection between graph theory and universal algebra. This makes it possible to investigate graph varieties and graph quasivarieties, i.e., classes of graphs described by identities or quasi-identities. In this paper, graph quasivarieties are characterized as classes of graphs closed under directed unions of isomorphic copies of finite strong pointed subproducts.

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