arXiv subjects

Ivan Chajda

Publications and source records attributed to Ivan Chajda.

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

Lattices and semilattices derived from commutative rings of characteristic 2 satisfying the identity $x^{2^n}\approx x$

We prove that a commutative ring $\mathbf R=(R,+,\cdot)$ of characteristic $2$ satisfying the identity $x^{2^n}\approx x$ together with the binary relation $\le$ on $R$ defined by $x\le y$ if $xy=x^2$ forms a meet-semilattice with smallest element $0$. If, moreover, $\mathbf R$ is unitary then we derive two binary term operations $\wedge$ and $\vee$ on $R$ which together with the unary term operation $x':=x+1$ form a Boolean algebra.

math.RA↗

arXiv2026

Residuated mappings and homomorphisms in posets

The concept of weakly regular residuated mappings was investigated for lattices recently by S. Radeleczki and L. Veres. We modify this concept for posets. We define so-called operator-residuated mappings in posets, show their important properties and point out differences between them and residuated mappings defined in the usual way. We modify the concept of a lattice homomorphism for posets where suprema need not exist and show its relation to residuated mappings and induced ideals. Finally, we present several examples of weakly regular residuated mappings in posets and show how such mappings may be constructed.

math.CO↗

arXiv2026

A Cayley theorem for posets

We show that every poset P=(P,\le) satisfying the Ascending Chain Condition can be isomorphically embedded into the poset of all mappings from P to the set A(P) of all antichains of P equipped with a certain partial order relation. This isomorphism is presented explicitly.

math.GM↗

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

Lagrange-like interpolation in unitary rings, Boolean algebras and Boolean posets

It is known that every function with a finite support over a given field can be interpolated by means of the Lagrangian polynomial. The question is if a similar interpolation is possible if one considers a unitary ring or a Boolean algebra instead of a field. We get a positive answer to this question provided the similarity type of the algebra in question is enriched with one more unary operation, the so-called Baaz delta. We get an explicit construction of this interpolation polynomial in both the cases. When going to Boolean posets, we have a lack of operations but these can be substituted by the operators Min U and Max L. Hence, we generalize also the Baaz delta for posets as an operator and then we can derive an explicit interpolation term constructed by means of these operators also for Boolean posets.

math.RA↗

arXiv2025

Properties of the symmetric difference in lattices with complementation

The symmetric difference in Boolean lattices can be defined in two different but equivalent forms. However, it can be introduced also in every bounded lattice with complementation where these two forms need not coincide. We study lattices with complementation and the variety of such lattices where these two expressions coincide and point out explicitly some interesting subvarieties. Using a result of J. Berman we estimate the size of free algebras in these subvarieties. It is well-known that the symmetric difference is associative in every Boolean lattice. We prove that it is just the property of Boolean lattices, namely the symmetric difference in a lattice with complementation is associative if and only if this lattice is Boolean. Similarly, we prove that a lattice with complementation is Boolean if and only if the symmetric difference satisfies a certain simple identity in two variables. We also characterize lattices with a unary operation satisfying De Morgan's laws.

math.RA↗

arXiv2025

Operators Max L and Min U and duals of Boolean posets

When working with posets which are not necessarily lattices, one has a lack of lattice operations which causes problems in algebraic constructions. This is the reason why we use the operators Max L and Min U substituting infimum and supremum, respectively. We axiomatize these operators. Two more operators, namely the so-called symmetric difference and the Sheffer operator, are introduced and studied in complemented posets by using the operators Max L and Min U. In Boolean algebras, the symmetric difference is used to construct its dual structure, the corresponding unitary Boolean ring. By generalizing this idea, we assign to each Boolean poset a so-called dual and prove that also, conversely, a Boolean poset can be derived from its dual.

math.LO↗

arXiv2025

Congruences on posets, relatively pseudocomplemented and Boolean posets

The aim of the present paper is to extend the concept of a congruence from lattices to posets. We use an approach different from that used by the first author and V. Sn\'a\v{s}el. By using our definition we show that congruence classes are convex. If the poset in question satisfies the Ascending Chain Condition as well as the Descending Chain Condition, then these classes turn out to be intervals. If the poset has a top element 1 then the 1-class of every congruence is a so-called strong filter. We study congruences on relatively pseudocomplemented posets which form a formalization of intuitionistic logic. For such posets we define so-called deductive systems and we show how they are connected with congruence kernels. We prove that every strong filter F of a relatively pseudocomplemented poset induces a congruence having F as its kernel. Finally, we consider Boolean posets which form a natural generalization of Boolean algebras. We show that congruences on Boolean posets in general do not share properties known from Boolean algebras, but congruence kernels of Boolean posets still have some interesting properties.

math.CO↗

arXiv2025

Derived operators on skew orthomodular and strong skew orthomodular posets

It is well-known that in the logic of quantum mechanics disjunctions and conjunctions can be represented by joins and meets, respectively, in an orthomodular lattice provided their entries commute. This was the reason why J. Pykacz introduced new derived operations called ''sharp'' and ''flat'' coinciding with joins and meets, respectively, for commuting elements but sharing some appropriate properties with disjunction and conjunction, respectively, in the whole orthomodular lattice in question. The problem is that orthomodular lattices need not formalize the logic of quantum mechanics since joins may not be defined provided their entries are neither comparable nor orthogonal. A corresponding fact holds for meets. Therefore, orthomodular posets are more accepted as an algebraic formalization of such a logic. The aim of the present paper is to extend the concepts of ''sharp'' and ''flat'' operations to operators in skew orthomodular and strong skew orthomodular posets. We generalize the relation of commuting elements as well as the commutator to such posets and we present some important properties of these operators and their mutual relationships. Moreover, we show that if the poset in question is even Boolean then there can be defined a ternary operator sharing the identities of a Pixley term. Finally, under some weak conditions which are automatically satisfied in Boolean algebras we show some kind of adjointness for operators formalizing conjunction and implication, respectively.

math.RA↗

arXiv2024

The variety of complemented lattices where the Sasaki operations form an adjoint pair

The Sasaki projection was introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. To use this projection and its dual so-called Sasaki operations were introduced by the second two authors. In a previous paper there are described several classes of lattices, $\lambda$-lattices and semirings where the Sasaki operations form an adjoint pair. In the present paper we prove that the class of complemented lattices with this property forms a variety and we explicitly state its defining identities. Moreover, we prove that this variety V is congruence permutable and regular. Hence every ideal I of some member L of V is a kernel of some congruence on L. Finally, we determine a finite basis of so-called ideal terms and describe the congruence $\Theta_I$ determined by the ideal I.

math.LO↗

arXiv2024

Adjointness of generalized Sasaki operations in posets

The Sasaki projection and its dual were introduced as a mapping from the lattice of closed subspaces of a Hilbert space onto one of its segments. In a previous paper the authors showed that the Sasaki operations induced by the Sasaki projection and its dual form an adjoint pair in every orthomodular lattice. Later on the authors described large classes of algebras in which Sasaki operations can be defined and form an adjoint pair. The aim of the present paper is to extend these investigations to bounded posets with a unary operation. We introduce the so-called generalized Sasaki projection and its dual as well as the so-called generalized Sasaki operations induced by them. When treating these projections and operations we consider only so-called saturated posets, i.e. posets having the property that above any lower bound of two elements there is at least one maximal lower bound and below any upper bound of two elements there is at least one minimal upper bound. We prove that the generalized Sasaki operations are well-defined if and only if the poset in question is orthogonal. We characterize adjointness of the generalized Sasaki operations in different ways and show that adjointness is possible only if the unary operation is a complementation. Finally, we prove that in every saturated orthomodular poset the generalized Sasaki operations form an adjoint pair.

math.RA↗

arXiv2024

Quasimodules over bounded lattices

We define a quasimodule Q over a bounded lattice L in an analogous way as a module over a semiring is defined. The essential difference is that L need not be distributive. Also for quasimodules there can be introduced the concepts of inner product, orthogonal elements, orthogonal subsets, bases and closed subquasimodules. We show that the set of all closed subquasimodules forms a complete lattice having orthogonality as an antitone involution. Using the Galois connection induced by this orthogonality, we describe important properties of closed subquasimodules. We call a subquasimodule P of a quasimodule Q splitting if the sum of P and its orthogonal companion is the whole set Q and the intersection of P and its orthogonal companion is trivial. We show that every splitting subquasimodule is closed and that its orthogonal companion is splitting, too. Our results are illuminated by several examples.

math.RA↗

arXiv2024

The operator of relative complementation

By the operator of relative complementation is meant a mapping assigning to every element x of an interval [a,b] of a lattice L the set x^{ab} of all relative complements of x in [a,b]. Of course, if L is relatively complemented then x^{ab} is non-empty for each interval [a,b] and every element x belonging to it. We study the question under what condition a complement of x in L induces a relative complement of x in [a,b] It is well-known that this is the case provided L is modular and complemented. However, we present a more general result. Further, we investigate properties of the operator of relative complementation, in particular in the case when the interval [a,b] is a modular sublattice of L or if it is finite. Moreover, we characterize when the operator of relative complementation is involutive and we show a class of lattices where this identity holds. Finally, we establish sufficient conditions under which two different complements of a given element x of [a,b] induce the same relative complement of x in this interval.

math.RA↗

arXiv2024

Algebras and varieties where Sasaki operations form an adjoint pair

The so-called Sasaki projection was introduced by U. Sasaki on the lattice L(H) of closed linear subspaces of a Hilbert space H as a projection of L(H) onto a certain sublattice of L(H). Since L(H) is an orthomodular lattice, the Sasaki projection and its dual can serve as the logical connectives conjunction and implication within the logic of quantum mechanics. It was shown by the authors in a previous paper that these operations form a so-called adjoint pair. The natural question arises if this result can be extended also to lattices with a unary operation which need not be orthomodular or to other algebras with two binary and one unary operation. To show that this is possible is the aim of the present paper. We determine a variety of lattices with a unary operation where the Sasaki operations form an adjoint pair and we continue with so-called $\lambda$-lattices and certain classes of semirings. We show that the Sasaki operations have a deeper sense than originally assumed by their author and can be applied also outside the lattices of closed linear subspaces of a Hilbert space.

math.RA↗

arXiv2024

Tense logics based on posets

Not all logical systems can be captured using algebras. We see this in classical logic (formalized by Boolean algebras) and many-valued logics (like Lukasiewicz logic with MV-algebras). Even quantum mechanics, initially formalized with orthomodular lattices, benefits from a simpler approach using just partially ordered sets (posets). This paper explores how logical connectives are introduced in poset-based logics. Building on prior work by the authors, we delve deeper into "dynamic" logics where truth values can change over time. We consider time sets with a preference relation and propositions whose truth depends on time. Tense operators, introduced by J.Burgess and extended for various logics, become a valuable tool. This paper proposes several approaches to this topic, aiming to inspire a further stream of research.

math.LO↗

arXiv2024

Operators on complemented lattices

The present paper deals with complemented lattices where, however, a unary operation of complementation is not explicitly assumed. This means that an element can have several complements. The mapping $^+$ assigning to each element $a$ the set $a^+$ of all its complements is investigated as an operator on the given lattice. We can extend the definition of $a^+$ in a natural way from elements to arbitrary subsets. In particular we study the set $a^+$ for complemented modular lattices, and we characterize when the set $a^{++}$ is a singleton. By means of the operator $^+$ we introduce two other operators $\to$ and $\odot$ which can be considered as implication and conjunction in a certain propositional calculus, respectively. These two logical connectives are ``unsharp'' which means that they assign to each pair of elements a non-empty subset. However, also these two derived operators share a lot of properties with the corresponding logical connectives in intuitionistic logic or in the logic of quantum mechanics. In particular, they form an adjoint pair. Finally, we define so-called deductive systems and we show their relationship to the mentioned operators as well as to lattice filters.

math.LO↗

arXiv2024

Induced orthogonality in semilattices with 0 and in pseudocomplemented lattices and posets

On an arbitrary meet-semilattice S with 0 we define an orthogonality relation and investigate the lattice Cl(S) of all subsets of S closed under this orthogonality. We show that if S is atomic then Cl(S) is a complete atomic Boolean algebra. If S is a pseudocomplemented lattice, this orthogonality relation can be defined by means of the pseudocomplementation. Finally, we show that if S is a complete pseudocomplemented lattice then Cl(S) is a complete Boolean algebra. For pseudocomplemented posets a similar result holds if the subset of pseudocomplements forms a complete lattice satisfying a certain compatibility condition.

math.CO↗