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Gejza Jenča

Publications and source records attributed to Gejza Jenča.

At least 19 recordsLinked to original sources
arXiv2025

Coherent configurations and Frobenius structures

We prove that coherent configurations can be represented as modules over Frobenius structures in the category of real nonnegative matrices. We generalize the notion of admissible morphism from association schemes to coherent configurations. We show that the Frobenius structure associated to a coherent configuration can be modified to become a dagger Frobenius structure, and use this to connect the coherent configurations to groupoids and $H^*$-algebras. We examine the properties of the dagger Frobenius structure with respect to admissible morphisms. We introduce the matrix $O$ obtained as the composition of comultiplication and multiplication of the dagger Frobenius structure and prove that we may obtain the valencies of colors, and thus recover the original coherent configuration, as an eigenvector of $O$. In the last part of the paper, we examine the spectrum of $O$ and apply it to generalize the Lagrange theorem from groups to association schemes.

math.CO↗
arXiv2025

Monoidal Quantaloids

We investigate how to add a symmetric monoidal structure to quantaloids in a compatible way. In particular, dagger compact quantaloids turn out to have properties that are similar to the category Rel of sets and binary relations. Examples of such quantaloids are the category qRel of quantum sets and binary relations, and the category V-Rel of sets and binary relations with values in a commutative quantale V. For both examples, the process of internalization structures is of interest. Discrete quantization, a process of generalization of mathematical structures to the noncommutative setting can be regarded as the process of internalizing these structures in qRel, whereas fuzzification, the process of introducing degrees of truth or membership to concepts that are traditionally considered either true or false, can be regarded as the process of internalizing structures in V-Rel. Hence, we investigate how to internalize power sets and preordered structures in dagger compact quantaloids.

math.CT↗
arXiv2024

N-free posets and orthomodularity

We prove that the incomparability orthoset of a finite poset is Dacey if and only if the poset is N-free. We give a characterization of finite posets with compatible incomparability orthosets.

math.CO↗
arXiv2023

Quantum Suplattices

Building on the theory of quantum posets, we introduce a non-commutative version of suplattices, i.e., complete lattices whose morphisms are supremum-preserving maps, which form a step towards a new notion of quantum topological spaces. We show that the theory of these quantum suplattices resembles the classical theory: the opposite quantum poset of a quantum suplattice is again a quantum suplattice, and quantum suplattices arise as algebras of a non-commutative version of the monad of downward-closed subsets of a poset. The existence of this monad is proved by introducing a non-commutative generalization of monotone relations between quantum posets, which form a compact closed category. Moreover, we introduce a non-commutative generalization of Galois connections and we prove that an upper Galois adjoint of a monotone map between quantum suplattices exists if and only if the map is a morphism of quantum suplattices. Finally, we prove a quantum version of the Knaster-Tarski fixpoint theorem: the quantum set of fixpoints of a monotone endomap on a quantum suplattice form a quantum suplattice.

cs.DM↗
arXiv2022

Orthogonality spaces associated with posets

An orthogonality space is a set equipped with a symmetric, irreflexive relation called orthogonality. Every orthogonality space has an associated complete ortholattice, called the logic of the orthogonality space. To every poset, we associate an orthogonality space consisting of proper quotients (that means, nonsingleton closed intervals), equipped with a certain orthogonality relation. We prove that a finite bounded poset is a lattice if and only if the logic of its orthogonality space is an orthomodular lattice. We prove that that a poset is a chain if and only if the logic of the associated orthogonality space is a Boolean algebra.

math.RA↗
arXiv2018

Two monads on the category of graphs

We introduce two monads on the category of graphs and prove that their Eilenberg-Moore categories are isomorphic to the category of perfect matchings and the category of partial Steiner triple systems, respectively. As a simple application of these results, we describe the product in the categories of perfect matchings and partial Steiner triple systems.

math.CO↗
arXiv2017

Effect algebras as presheaves on finite Boolean algebras

For an effect algebra $A$, we examine the category of all morphisms from finite Boolean algebras into $A$. This category can be described as a category of elements of a presheaf $R(A)$ on the category of finite Boolean algebras. We prove that some properties (being an orthoalgebra, the Riesz decomposition property, being a Boolean algebra) of an effect algebra $A$ can be characterized by properties of the category of elements of the presheaf $R(A)$. We prove that the tensor product of of effect algebras arises as a left Kan extension of the free product of finite Boolean algebras along the inclusion functor. As a consequence, the tensor product of effect algebras can be expressed by means of the Day convolution of presheaves on finite Boolean algebras.

math.RA↗
arXiv2017

A note on unitizations of generalized effect algebras

There is a forgetful functor from the category of generalized effect algebras to the category of effect algebras. We prove that this functor is a right adjoint and that the corresponding left adjoint is the well-known unitization construction by Hedlíková and Pulmannová. Moreover, this adjunction is monadic.

math.RA↗
arXiv2017

On monoids in the category of sets and relations

The category $\mathbf{Rel}$ is the category of sets (objects) and relations (morphisms). Equipped with the direct product of sets, $\mathbf{Rel}$ is a monoidal category. Moreover, $\mathbf{Rel}$ is a locally posetal 2-category, since every homset $\mathbf{Rel}(A,B)$ is a poset with respect to inclusion. We examine the 2-category of monoids $\mathbf{RelMon}$ in this category. The morphism we use are lax. This category includes, as subcategories, various interesting classes: hypergroups, partial monoids (which include various types of quantum logics, for example effect algebras) and small categories. We show how the 2-categorical structure gives rise to several previously defined notions in these categories, for example certain types of congruence relations on generalized effect algebras. This explains where these definitions come from.

math.RA↗
arXiv2015

Linear extensions and order-preserving poset partitions

We examine the lattice of all order congruences of a finite poset from the viewpoint of combinatorial algebraic topology. We will prove that the order complex of the lattice of all nontrivial order congruences (or order-preserving partitions) of a finite $n$-element poset $P$ with $n\geq 3$ is homotopy equivalent to a wedge of spheres of dimension $n-3$. If $P$ is connected, then the number of spheres is equal to the number of linear extensions of $P$. In general, the number of spheres is equal to the number of cyclic extensions of $P$.

math.CO↗
arXiv2015

Blocks of homogeneous effect algebras

Effect algebras, introduced by Foulis and Bennett in 1994, are partial algebras which generalize some well known classes of algebraic structures (for example orthomodular lattices, MV algebras, orthoalgebras etc.). In the present paper, we introduce a new class of effect algebras, called {\em homogeneous effect algebras}. This class includes orthoalgebras, lattice ordered effect algebras and effect algebras satisfying Riesz decomposition property. We prove that every homogeneous effect algebra is a union of its blocks, which we define as maximal sub-effect algebras satisfying Riesz decomposition property. This generalizes a recent result by Riečanová, in which lattice ordered effect algebras were considered. Moreover, the notion of a block of a homogeneous effect algebra is a generalization of the notion of a block of an orthoalgebra. We prove that the set of all sharp elements in a homogeneous effect algebra $E$ forms an orthoalgebra $E_S$. Every block of $E_S$ is the center of a block of $E$. The set of all sharp elements in the compatibility center of $E$ coincides with the center of $E$. Finally, we present some examples of homogeneous effect algebras and we prove that for a Hilbert space $\mathbb H$ with $dim(\mathbb H)>1$, the standard effect algebra $\mathcal E(\mathbb H)$ of all effects in $\mathbb H$ is not homogeneous.

math.RA↗
arXiv2010

Extensions of witness mappings

We deal with the problem of coexistence in interval effect algebras using the notion of a witness mapping. Suppose that we are given an interval effect algebra $E$, a coexistent subset $S$ of $E$, a witness mapping $β$ for $S$, and an element $t\in E\setminus S$. We study the question whether there is a witness mapping $β_t$ for $S\cup\{t\}$ such that $β_t$ is an extension of $β$. In the main result, we prove that such an extension exists if and only if there is a mapping $e_t$ from finite subsets of $S$ to $E$ satisfying certain conditions. The main result is then applied several times to prove claims of the type "If $t$ has a such-and-such relationship to $S$ and $β$, then $β_t$ exists".

math-ph↗