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Michal Botur

Publications and source records attributed to Michal Botur.

At least 19 recordsLinked to original sources
arXiv2026

Orthogonal adjointness in posets with $0$

Motivated by the concept of polarity introduced by G. Birkhoff for a binary relation on a set, we introduce a concept of orthogonality in a poset with $0$. A pair of operators $f$, $g$ on a poset with $0$ is called orthogonally adjoint if $f(x)$ is orthogonal to $y$ if and only if $x$ is orthogonal to $g(y)$. We characterize the existence and the uniqueness of $g$ for given $f$ and describe basic properties of orthogonal adjointness. We present constructions of orthogonally adjoint pairs in pseudocomplemented posets. If a given operator $f$ is an order-isomorphism of a pseudocomplemented poset satisfying some natural properties then the corresponding adjoint $g$ can be described explicitly. Moreover, if $f$ and $f^{-1}$ are bijective $\perp$-morphisms then they are orthogonally adjoint, too. Finally we show that a given pair of orthogonally adjoint mappings on a poset $\mathbf P$ may not be extendable to the Dedekind-McNeille completion of $\mathbf P$ and we present sufficient conditions for the existence of such an extension. We also provide sufficient conditions for the existence of an extension of orthogonally adjoint mappings to the lattice of ideals. Our results are illustrated by numerous examples.

math.CO↗
arXiv2026

Quantales carrying ortholattice structure

This paper investigates the intersection of residuated structures from many-valued logic and orthomodular lattices from quantum logic. We explore whether non-Boolean structures can simultaneously satisfy residuation principles and orthocomplementation requirements. Our main contribution is a study of Girard posets with inversions, providing a characterization theorem where a unital residuated poset is Girard if and only if it admits an inversion satisfying specific adjointness conditions. We prove that any complemented lattice admitting an integral residuated structure must be Boolean, which motivates our search for orthomodular examples in the non-integral case. We answer this by demonstrating that the lattice $C(\mathbb{R}^n)$ of closed subspaces of $n$-dimensional real coordinate space carries both an orthomodular and a commutative Girard quantale structure. This construction provides a concrete non-Boolean framework unifying quantum-logical and many-valued logical reasoning.

math.LO↗
arXiv2025

A new representation of finite Hoops using a new type of product of structures

In this paper we show that a new type of products hoops can be defined which, in the case of finite hoops, can describe an arbitrary hoop $\mathbf A$ as the product of its arbitrary filter $F$ and the corresponding homomorphic image $\mathbf A/F$. Moreover, this product satisfies a certain kind of associativity, and as a consequence we show that every finite hoop is in this sense a product of finite MV-chains.

math.LO↗
arXiv2025

Operators on complemented posets

Given a complemented poset P, we can assign to every element x of P the set x^+ of all its complements. We study properties of the operator ^+ on P, in particular, we are interested in the case when x^+ forms an antichain or when ^+ is involutive or antitone. We apply ^+ to the set Min U(x,y) of all minimal elements of the upper cone U(x,y) of x,y and to the set Max L(x,y) of all maximal elements of the lower cone L(x,y) of x,y. By using ^+ we define four binary operators on P and investigate their properties that are close to adjointness. We present an example of a uniquely complemented poset that is not Boolean. In the last section we study the orthogonality relation induced by complementation. We characterize when two elements of the Dedekind-MacNeille completion of P are orthogonal to each other. Finally, we extend the orthogonality relation from elements to subsets and we prove that two non-empty subsets of P are orthogonal to each other if and only if their convex hulls are orthogonal to each other within the poset of all non-empty convex subsets of P.

math.LO↗
arXiv2025

Beyond wreath and block

We investigate a semigroup construction generalising the two-sided wreath product. We develop the foundations of this construction and show that for groups it is isomorphic to the usual wreath product. We also show that it gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop monoid is replaced by the two-element semilattice.

math.GR↗
arXiv2025

Kites and representations of pseudo MV-algebras

We investigate the structure of perfect residuated lattices, focussing especially on perfect pseudo MV-algebras. We show that perfect pseudo MV-algebras can be represented as a generalised version of kites of Dvure\v{c}enskij and Kowalski, and that they are categorically equivalent to $\ell$-groups with a distinguished automorphism. We then characterise varieties generated by kites and describe the lattice of these varieties as a complete sublattice of the lattice of perfectly generated varieties of perfect pseudo MV-algebras.

math.LO↗
arXiv2025

Many-valued aspects of tense an related operators

Our research builds upon Halmos's foundational work on functional monadic Boolean algebras and our previous work on tense operators to develop three essential constructions, including the important concepts of fuzzy sets and powerset operators. These constructions have widespread applications across contemporary mathematical disciplines, including algebra, logic, and topology. The framework we present generates four covariant and two contravariant functors, establishing three adjoint situations.

math.LO↗
arXiv2025

Foulis quantales and complete orthomodular lattices

Our approach establishes a natural correspondence between complete orthomodular lattices and certain types of quantales. Firstly, given a complete orthomodular lattice X, we associate with it a Foulis quantale Lin(X) consisting of its endomorphisms. This allows us to view X as a left module over Lin(X), thereby introducing a novel fuzzy-theoretic perspective to the study of complete orthomodular lattices. Conversely, for any Foulis quantale Q, we associate a complete orthomodular lattice [Q] that naturally forms a left Q-module. Furthermore, there exists a canonical homomorphism of Foulis quantales from Q to Lin([Q]).

math.LO↗
arXiv2025

A dagger kernel category of complete orthomodular lattices

Dagger kernel categories, a powerful framework for studying quantum phenomena within category theory, provide a rich mathematical structure that naturally encodes key aspects of quantum logic. This paper focuses on the category SupOMLatLin of complete orthomodular lattices with linear maps. We demonstrate that SupOMLatLin itself forms a dagger kernel category, equipped with additional structure such as dagger biproducts and free objects. A key result establishes that every morphism in SupOMLatLin admits an essentially unique factorization as a zero-epi followed by a dagger monomorphism. This factorization theorem, along with the dagger kernel category structure of SupOMLatLin, provides new insights into the interplay between complete orthomodular lattices and the foundational concepts of quantum theory.

math.LO↗
arXiv2025

Foulis m-semilattices and their modules

Building upon the results of Jacobs, we show that the category OMLatLin of orthomodular lattices and linear maps forms a dagger category. For each orthomodular lattice X, we construct a Foulis m-semilattice Lin(X) composed of endomorphisms of X. This m-semilattice acts as a quantale, enabling us to regard X as a left Lin(X)-module. Our novel approach introduces a fuzzy-theoretic dimension to the theory of orthomodular lattices.

math.LO↗
arXiv2022

Another look on tense and related operators

Motivated by the classical work of Halmos on functional monadic Boolean algebras we derive three basic sup-semilattice constructions, among other things the so-called powersets and powerset operators. Such constructions are extremely useful and can be found in almost all branches of modern mathematics, including algebra, logic and topology. Our three constructions give rise to four covariant and two contravariant functors and constitute three adjoint situations we illustrate in simple examples.

math.RA↗
arXiv2018

On generating of idempotent aggregation functions on finite lattices

In a recent paper we proposed the study of aggregation functions on lattices via clone theory approach. Observing that aggregation functions on lattices just correspond to $0,1$-monotone clones, we have shown that all aggregation functions on a finite lattice $L$ can be obtained as usual composition of lattice operations $\wedge,\vee$, and certain unary and binary aggregation functions. The aim of this paper is to present a generating set for the class of intermediate (or, equivalently, idempotent) aggregation functions. This set consists of lattice operations and certain ternary idempotent aggregation functions.

math.RA↗
arXiv2018

The Conrad Program: From l-groups to algebras of logic

A number of research articles have established the significant role of lattice-ordered groups (l-groups) in logic. The purpose of the present article is to lay the groundwork for, and provide significant initial contributions to, the development of a Conrad type approach to the study of algebras of logic. The term Conrad Program refers to Paul Conrad's approach to the study of l-groups, which analyzes the structure of individual l-groups or classes of l-groups by primarily using strictly lattice theoretic properties of their lattices of convex l-subgroups. The present article demonstrates that large parts of the Conrad Program can be profitably extended in the setting of e-cyclic residuated lattices. An indirect benefit of this work is the introduction of new tools and techniques in the study of algebras of logic, and the enhanced role of the lattice of convex subalgebras of a residuated lattice.

math.RA↗
arXiv2018

Filters on some classes of quantum B-algebras

In this paper, we continue the study of quantum B-algebras with emphasis on filters on integral quantum B-algebras. We then study filters in the setting of pseudo-hoops. First, we establish an embedding of a cartesion product of polars of a pseudo-hoop into itself. Second, we give sufficient conditions for a pseudohoop to be subdirectly reducible. We also extend the result of Kondo and Turunen to the setting of noncommutative residuated $\vee$-semilattices that, if prime filters and $\vee$-prime filters of a residuated $\vee$-semilattice $A$ coincide, then $A$ must be a pseudo MTL-algebra.

math.LO↗
arXiv2018

Two-sided wreath product done right

We investigate a semigroup construction related to the two-sided wreath product. It encompasses a range of known constructions and gives a slightly finer version of the decomposition in the Krohn-Rhodes Theorem, in which the three-element flip-flop is replaced by the two-element semilattice. We develop foundations of the theory of our construction, showing in the process that it naturally combines ideas from semigroup theory (wreath products), category theory (Grothendieck construction), and ordered structures (residuated lattices).

math.RA↗
arXiv2017

Kites and Residuated Lattices

We investigate a construction of an integral residuated lattice starting from an integral residuated lattice and two sets with an injective mapping from one set into the second one. The resulting algebra has a shape of a Chinese cascade kite, therefore, we call this algebra simply a kite. We describe subdirectly irreducible kites and we classify them. We show that the variety of integral residuated lattices generated by kites is generated by all finite-dimensional kites. In particular, we describe some homomorphisms among kites.

math.RA↗
arXiv2014

Kite $n$-Perfect Pseudo Effect Algebras

Kite pseudo effect algebras were recently introduced as a class of interesting examples of pseudo effect algebras using a po-group, an index set and two bijections on the index set. We represent kite pseudo effect algebras with a special kind of the Riesz decomposition property as an interval in a lexicographic extension of the po-group which solves an open problem on representation of kites. In addition, we introduce kite $n$-perfect pseudo effect algebras and we characterize subdirectly irreducible algebras which are building stones of the theory.

math.RA↗