arXiv subjects

Ivan Chajda

Publications and source records attributed to Ivan Chajda.

At least 37 records · Page 2Linked to original sources

arXiv2024

A characterization of permutability of 2-uniform tolerances on posets

Tolerance relations were investigated by several authors in various algebraic structures, see e.g. the monograph by I. Chajda. Recently G. Czédli studied so-called 2-uniform tolerances on lattices, i.e. tolerances that are compatible with the lattice operations and whose blocks are of cardinality 2. He showed that two such tolerances on a lattice containing no infinite chain permute if and only if they are amicable (a concept introduced in his paper). We extend this study to tolerances on posets. Since in posets we have no lattice operations, we must modify the notion of amicability. We modified it in such a way that in case of lattices it coincides with the original definition. With this new definition we can prove that two tolerances on a poset containing no infinite chain permute if and only if they are amicable in the new sense.

math.CO↗

arXiv2024

Implications in pseudocomplemented and Stone lattices

Two kinds of the connective implication are introduced as term operations of a pseudocomplemented lattice. It is shown that they share a lot of properties with the intuitionistic implication based on Heyting algebras. In particular, if the pseudocomplemented lattice in question is a Stone lattice then the considered implications satisfy some kind of quasi-commutativity, of the exchange property, some version of adjointness with the meet-operation and some kind of the derivation rule Modus Ponens and of the contraposition law. Two kinds of deductive systems are defined and their elementary properties are shown. All investigated concepts are illuminated by examples.

math.LO↗

arXiv2023

Properties and special filters of pseudocomplemented posets

Investigating the structure of pseudocomplemented lattices started ninety years ago with papers by V. Glivenko, G. Birkhoff and O. Frink and this structure was essentially developed by G. Grätzer. In recent years, some special filters in pseudocomplemented and Stone lattices have been studied by M. Sambasiva Rao. However, in some applications, in particular in non-classical logics with unsharp logical connectives, pseudocomplemented posets instead of lattices are used. This motivated us to develop an algebraic theory of pseudocomplemented posets, i.e. we derive identities and inequalities holding in such posets and we use them in order to characterize the so-called Stone posets. Then we adopt several concepts of special filters and we investigate their properties in pseudocomplemented posets. Moreover, we show how properties of these filters influence algebraic properties of the underlying pseudocomplemented posets.

math.RA↗

arXiv2023

The intuitionistic-like logic based on a poset

The aim of the present paper is to show that the concept of intuitionistic logic based on a Heyting algebra can be generalized in such a way that it is formalized by means of a bounded poset. In this case it is not assumed that the poset is relatively pseudocomplemented. The considered logical connectives negation, implication or even conjunction are not operations in this poset but so-called operators since they assign to given entries not necessarily an element of the poset as a result but a subset of mutually incomparable elements with maximal possible truth values. We show that these operators for negation and implication can be characterized by several simple conditions formulated in the language of posets together with the operator of taking the lower cone. Moreover, our implication and conjunction form an adjoint pair. We call these connectives "unsharp" or "inexact" in accordance with the existing literature. We also introduce the concept of a deductive system of a bounded poset with implication and prove that it induces an equivalence relation satisfying a certain substitution property with respect to implication. Moreover, the restriction of this equivalence on the base set is uniquely determined by its kernel, i.e. the class containing the top element.

math.LO↗

arXiv2023

Special filters in bounded lattices

M.S. Rao recently investigated some sorts of special filters in distributive pseudocomplemented lattices. In our paper we extend this study to lattices which need neither be distributive nor pseudocomplemented. For this sake we define a certain modification of the notion of a pseudocomplement as the set of all maximal elements belonging to the annihilator of the corresponding element. We prove several basic properties of this notion and then define coherent, closed and median filters as well as D-filters. In order to be able to obtain valuable results we often must add some additional assumptions on the underlying lattice, e.g. that this lattice is Stonean or D-Stonean. Our results relate properties of lattices and of corresponding filters. We show how the structure of a lattice influences the form of its filters and vice versa.

math.RA↗

arXiv2023

The logic of effect algebras incorporating time dimension

Effect algebras were introduced in order to describe the structure of effects, i.e. events in quantum mechanics. They are partial algebras describing the logic behind the corresponding events. It is natural to ask how to introduce the logical connective implication in effect algebras. For lattice-ordered effect algebras this task was already solved by several authors, including the present ones. We concentrate on effect algebras that need not be lattice-ordered since these can better describe the events occurring in the quantum physical system. Although an effect algebra is only partial, we try to find a logical connective implication which is everywhere defined. But such a connective can be "unsharp" or "inexact" because its outputs for given pairs of entries need not be elements of the underlying effect algebra, but may be subsets of (mutually incomparable) maximal elements. We introduce such an implication together with its adjoint functor representing conjunction. Then we consider so-called tense operators on effect algebras. Of course, also these operators turn out to be "unsharp" in the aforementioned sense, but they are in a certain relation with the operators implication and conjunction. Finally, for given tense operators and given time set T, we describe two methods how to construct a time preference relation R on T such that the given tense operators are either comparable with or equivalent to those induced by the time frame (T,R).

math.LO↗

arXiv2023

Implication in sharply paraorthomodular and relatively paraorthomodular posets

In this paper we show that several classes of partially ordered structures having paraorthomodular reducts, or whose sections may be regarded as paraorthomodular posets, admit a quite natural notion of implication, that admits a suitable notion of adjointness. Within this framework, we propose a smooth generalization of celebrated Greechie's theorems on amalgams of finite Boolean algebras to the realm of Kleene lattices.

math.LO↗

arXiv2023

The logic with unsharp implication and negation

It is well-known that intuitionistic logics can be formalized by means of Brouwerian semilattices, i.e. relatively pseudocomplemented semilattices. Then the logical connective implication is considered to be the relative pseudocomplement and conjunction is the semilattice operation meet. If the Brouwerian semilattice has a bottom element 0 then the relative pseudocomplement with respect to 0 is called the pseudocomplement and it is considered as the connective negation in this logic. Our idea is to consider an arbitrary meet-semilattice with 0 satisfying only the Ascending Chain Condition, which is trivially satisfied in finite semilattices, and introduce the connective negation x^0 as the set of all maximal elements z satisfying x^z=0 and the connective implication x->y as the set of all maximal elements z satisfying x^z\le y. Such a negation and implication is "unsharp" since it assigns to one entry x or to two entries x and y belonging to the semilattice, respectively, a subset instead of an element of the semilattice. Surprisingly, this kind of negation and implication, respectively, still shares a number of properties of these connectives in intuitionistic logic, in particular the derivation rule Modus Ponens. Moreover, unsharp negation and unsharp implication can be characterized by means of five, respectively seven simple axioms. Several examples are presented. The concepts of a deductive system and of a filter are introduced as well as the congruence determined by such a filter. We finally describe certain relationships between these concepts.

math.LO↗

arXiv2022

Representability of Kleene posets and Kleene lattices

A Kleene lattice is a distributive lattice equipped with an antitone involution and satisfying the so-called normality condition. These lattices were introduced by J. A. Kalman. We extended this concept also for posets with an antitone involution. In our recent paper [5], we showed how to construct such Kleene lattices or Kleene posets from a given distributive lattice or poset and a fixed element of this lattice or poset by using the so-called twist product construction, respectively. We extend this construction of Kleene lattices and Kleene posets by considering a fixed subset instead of a fixed element. Moreover, we show that in some cases, this generating poset can be embedded into the resulting Kleene poset. We investigate the question when a Kleene poset can be represented by a Kleene poset obtained by the mentioned construction. We show that a direct product of representable Kleene posets is again representable and hence a direct product of finite chains is representable. This does not hold in general for subdirect products, but we show some examples where it holds. We present large classes of representable and non-representable Kleene posets. Finally, we investigate two kinds of extensions of a distributive poset A, namely its Dedekind-MacNeille completion DM(A) and a completion G(A) which coincides with DM(A) provided A is finite. In particular we prove that if A is a Kleene poset then its extension G(A) is also a Kleene lattice. If the subset X of principal order ideals of A is involutionclosed and doubly dense in G(A) then it generates G(A) and it is isomorphic to A itself.

math.LO↗

arXiv2022

The logic of quantum mechanics incorporating time dimension

Similarly as classical propositional calculus is based algebraically on Boolean algebras, the logic of quantum mechanics was based on orthomodular lattices by G. Birkhoff and J. von Neumann and K. Husimi. However, this logic does not incorporate time dimension although it is apparent that the propositions occurring in the logic of quantum mechanics are depending on time. The aim of the present paper is to show that so-called tense operators can be introduced also in such a logic for given time set and given time preference relation. In this case we can introduce these operators in a purely algebraic way. We derive several important properties of such operators, in particular we show that they form dynamic pairs and, altogether, a dynamic algebra. We investigate connections of these operators with logical connectives conjunction and implication derived from Sasaki projections. Then we solve the converse problem, namely to find for given time set and given tense operators a time preference relation in order that the resulting time frame induces the given operators. We show that the given operators can be obtained as restrictions of operators induced by a suitable extended time frame.

math.LO↗

arXiv2022

Orthomodular and generalized orthomodular posets

We prove that the 18-element non-lattice orthomodular poset depicted in the paper is the smallest one and unique up to isomorphism. Since not every Boolean poset is orthomodular, we consider the class of the so-called generalized orthomodular posets introduced by the first and third author in a previous paper. We show that this class contains all Boolean posets and we study its subclass consisting of horizontal sums of Boolean posets. For this purpose we introduce the concept of a compatibility relation and the so-called commutator of two elements. We show the relationship between these concepts and we introduce the notion of a ternary discriminator for these posets. Numerous examples illuminating these concepts and results are included in the paper.

math.QA↗

arXiv2022

Tense logic based on finite orthomodular posets

It is widely accepted that the logic of quantum mechanics is based on orthomodular posets. However, such a logic is not dynamic in the sense that it does not incorporate time dimension. To fill this gap, we introduce certain tense operators on such a logic in an inexact way, but still satisfying requirements asked on tense operators in the classical logic based on Boolean algebras or in various non-classical logics. Our construction of tense operators works perfectly when the orthomodular poset in question is finite. We investigate the behaviour of these tense operators, e.g. we show that some of them form a dynamic pair. Moreover, we prove that if the tense operators preserve one of the inexact connectives conjunction or implication as defined by the authors recently in another paper, then they also preserve the other one. Finally, we show how to construct the binary relation of time preference on a given time set provided the tense operators are given, up to equivalence induced by natural quasiorders.

math.LO↗

arXiv2022

Implication in weakly and dually weakly orthomodular lattices

Weakly orthomodular and dually weakly orthomodular lattices were introduced by the authors in a recent paper. Similarly as for orthomodular lattices we try to introduce an implication in these lattices which can be easily axiomatized and which yields a nice lattice structure. As shown in the paper, this can be realized in several different ways. Moreover, we reveal the connection of weakly and dually weakly orthomodular lattices to residuated structures. Moreover, we provide a characterization of these lattices by means of certain generalized measures.

math.RA↗

arXiv2022

c-ideals in complemented posets

In their recent paper on posets with a pseudocomplementation denoted by * the first and the third author introduced the concept of a *-ideal. This concept is in fact an extension of a similar concept introduced in distributive pseudocomplemented lattices and semilattices by several authors, see References. Now we apply this concept of a c-ideal (dually, c-filter) to complemented posets where the complementation need neither be antitone nor an involution, but still satisfies some weak conditions. We show when an ideal or filter in such a poset is a c-ideal or c-filter, respectively, and we prove basic properties of them. Finally, we prove so-called Separation Theorems for c-ideals. The text is illustrated by several examples.

math.RA↗

arXiv2022

Operator residuation in orthomodular posets of finite height

We show that for every orthomodular poset P of finite height there can be defined two operators forming an adjoint pair with respect to an order-like relation defined on the power set of P. This enables us to introduce the so-called operator residuated poset corresponding to P from which the original orthomodular poset P can be recovered. Moreover, this correspondence is almost one-to-one. We show that this construction of operators can be applied also to so-called weakly orthomodular and dually weakly orthomodular posets. Examples of such posets are included.

math.LO↗

arXiv2022

Remarks on special congruences

We study algebras and varieties where every non-trivial congruence has some class being a non-trivial subuniverse of the algebra in question. Then we focus on algebras where this non-trivial class is a unique non-singleton class of the congruence. In particular, we investigate Rees algebras, quasi-Rees algebras and algebras having the one-block-property. We also present results concerning these properties on quotient algebras. Several examples of such algebras and varieties are included.

math.RA↗

arXiv2022

Filters and ideals in pseudocomplemented posets

We study ideals and filters of posets and of pseudocomplemented posets and show a version of the Separation Theorem, known for ideals and filters in lattices and semilattices, within this general setting. We extend the concept of a *-ideal already introduced by Rao for pseudocomplemented distributive lattices and by Talukder, Chakraborty and Begum for pseudocomplemented semilattices to pseudocomplemented posets. We derive several important properties of such ideals. Especially, we explain connections between prime filters, ultrafilters, filters satisfying the *-condition and dense elements. Finally, we prove a Separation Theorem for *-ideals.

math.RA↗

arXiv2022

Extensions and congruences of partial lattices

For a partial lattice L the so-called two-point extension is defined in order to extend L to a lattice. We are motivated by the fact that the one-point extension broadly used for partial algebras does not work in this case, i.e. the one-point extension of a partial lattice need not be a lattice. We describe these two-point extensions and prove several properties of them. We introduce the concept of a congruence on a partial lattice and show its relationship to the notion of a homomorphism and its connections with congruences on the corresponding two-point extension. In particular we prove that the quotient L/E of a partial lattice L by a congruence E on L is again a partial lattice and that the two-point extension of L/E is isomorphic to the quotient lattice of the two-point extension L* of L by the congruence on L* generated by E. Several illustrative examples are enclosed.

math.RA↗