arXiv subjects

Ivan Chajda

Publications and source records attributed to Ivan Chajda.

At least 73 records · Page 4Linked to original sources

arXiv2020

Extensions of posets with an antitone involution to residuated structures

We prove that every not necessarily bounded poset P=(P,\le,') with an antitone involution can be extended to a residuated poset E(P)=(E(P),\le,\odot,\rightarrow,1) where x'=x\rightarrow0 for all x\in P. If P is a lattice with an antitone involution then E(P) is a lattice, too. We show that a poset can be extended to a residuated poset by means of a finite chain and that a Boolean algebra (B,\vee,\wedge,',p,q) can be extended to a residuated lattice (Q,\vee,\wedge,\odot,\rightarrow,1) by means of a finite chain in such a way that x\odot y=x\wedge y and x\rightarrow y=x'\vee y for all x,y\in B.

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arXiv2020

Sublattices and $Δ$-blocks of orthomodular posets

For orthoposets we introduce a binary relation Delta and a binary operator d(x,y) which are generalizations of the binary relation C and the commutator c(x,y), respectively, known for orthomodular lattices. We characterize orthomodular posets among orthoposets and orthogonal posets. Moreover, we describe connections between the relations Delta and \leftrightarrow and the operator d(x,y). In details we investigate certain orthomodular posets of subsets of a finite set. In particular we describe maximal orthomodular sublattices and Boolean subalgebras of such orthomodular posets. Finally, we study properties of Delta-blocks with respect to Boolean sublattices and distributive subposets they include.

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arXiv2020

The logic of orthomodular posets of finite height

Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. The question is how to introduce the connective implication in such a logic. We show that this is possible when the orthomodular poset in question is of finite height. The main point is that the corresponding algebra, called implication orthomodular poset, i.e. a poset equipped with a binary operator of implication, corresponds to the original orthomodular poset and this operator is everywhere defined. We present here the complete list of axioms for implication orthomodular posets. This enables us to derive an axiomatization in Gentzen style for the algebraizable logic of orthomodular posets of finite height.

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arXiv2020

Orthogonality and complementation in the lattice of subspaces of a finite-dimensional vector space over a finite field

We investigate the lattice L(V) of subspaces of an m-dimensional vector space V over a finite field GF(q) with q being the n-th power of a prime p. It is well-known that this lattice is modular and that orthogonality is an antitone involution. The lattice L(V) satisfies the Chain condition and we determine the number of covers of its elements, especially the number of its atoms. We characterize when orthogonality is a complementation and hence when L(V) is orthomodular. For m > 1 and m not divisible by p we show that L(V) contains a certain (non-Boolean) orthomodular lattice as a subposet. Finally, for q being a prime and m = 2 we characterize orthomodularity of L(V) by a simple condition.

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arXiv2020

The logic induced by effect algebras

Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras E we investigate a natural implication and prove that the implication reduct of E is term equivalent to E. Then we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the Ascending Chain Condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

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arXiv2019

Varieties corresponding to classes of complemented posets

As algebraic semantics of the logic of quantum mechanics there are usually used orthomodular posets, i.e. bounded posets with a complementation which is an antitone involution and where the join of orthogonal elements exists and the orthomodular law is satisfied. When we omit the condition that the complementation is an antitone involution, then we obtain skew-orthomodular posets. To each such poset we can assign a bounded lambda-lattice in a non-unique way. Bounded lambda-lattices are lattice-like algebras whose operations are not necessarily associative. We prove that any of the following properties for bounded posets with a unary operation can be characterized by certain identities of an arbitrary assigned lambda-lattice: complementarity, orthogonality, almost skew-orthomodularity and skew-orthomodularity. It is shown that these identities are independent. Finally, we show that the variety of skew-orthomodular lambda-lattices is congruence permutable as well as congruence regular.

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arXiv2019

Residuation in finite posets

When an algebraic logic based on a poset instead of a lattice is investigated then there is a natural problem how to introduce the connective implication to be everywhere defined and satisfying (left) adjointness with the connective conjunction. We have already studied this problem for the logic of quantum mechanics which is based on an orthomodular poset or the logic of quantum effects based on a so-called effect algebra which is only partial and need not be lattice-ordered. For this, we introduced the so-called operator residuation where the values of implication and conjunction need not be elements of the underlying poset, but only certain subsets of it. However, this approach can be generalized for posets satisfying more general conditions. If these posets are even finite, we can focus on maximal or minimal elements of the corresponding subsets and the formulas for the mentioned operators can be essentially simplified. This is shown in the present paper where all theorems are explained by corresponding examples.

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arXiv2019

Semimodular $λ$-lattices

The concept of a $λ$-lattice was introduced by V. Sná\v sel in order to generalize some lattice concepts for directed posets whose elements need not have suprema or infima. We extend the concept of semimodularity from lattices to $λ$-lattices and show connections to the lower covering condition and its generalizations. We further show that, contrary to the case of lattices, for $λ$-lattices semimodularity and the (weak) lower covering condition are independent properties. However, under some additional conditions semimodularity implies the (weak) lower covering condition. Examples of corresponding $λ$-lattices are presented.

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arXiv2019

Properties of the connective implication in effect algebras

Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serves as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is is equipped with a connective implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain "unsharp" residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

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arXiv2019

How to introduce the connective implication in orthomodular posets

Since orthomodular posets serve as an algebraic axiomatization of the logic of quantum mechanics, it is a natural question how the connective of implication can be defined in this logic. It should be introduced in such a way that it is related with conjunction, i.e. with the partial operation meet, by means of some kind of adjointness. We present here such an implication for which a so-called unsharp residuated poset can be constructed. Then this implication is connected with the operation meet by the so-called unsharp adjointness. We prove that also conversely, under some additional assumptions, such an unsharp residuated poset can be converted into an orthomodular poset and that this assignment is nearly one-to-one.

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arXiv2019

Semimodules over commutative semirings and modules over unitary commutative rings

We study the so-called closed and splitting subsemimodules and submodules of a given semimodule or module, respectively. We describe lattices of subsemimodules and of closed subsemimodules and posets of splitting subsemimodules and submodules. In the case of modules a natural bijective correspondence between these posets and posets of projections is established.

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arXiv2019

Unsharp residuation in effect algebras

Effect algebras and pseudoeffect algebras were introduced by Foulis, Bennett, Dvurecenskij and Vetterlein as so-called quantum structures which serve as an algebraic axiomatization of the logic of quantum mechanics. A natural question concerns their connections to substructural logics which are described by means of residuated lattices or posets. In a previous paper it was shown that an effect algebra can be organized into a so-called conditionally residuated structure where the adjointness condition holds only for those elements for which multiplication and implication are defined. Because this is a very strong restriction, we try to find another kind of residuation where the terms occurring in the adjointness condition are everywhere defined though the binary operation of a given effect algebra is only partial. Moreover, we work with effect algebras which need not be lattice-ordered and hence the lattice operations join and meet are replaced by means of upper and lower cones which, however, are not elements but subsets. Hence, the resulting concept, the so-called unsharp residuated poset, is equipped with LU-terms which substitute operations, but are everywhere defined. Although this concept seems rather complicated at the first glance, we prove that such an unsharp residuated poset can be conversely organized into an effect algebra or a pseudoeffect algebra depending on commutativity of the multiplication.

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arXiv2019

Sectionally pseudocomplemented posets

The concept of a sectionally pseudocomplemented lattice was introduced by I. Chajda as an extension of relative pseudocomplementation for not necessarily distributive lattices. The typical example of such a lattice is the non-modular lattice N5. The aim of this paper is to extend the concept of sectional pseudocomplementation from lattices to posets. At first we show that the class of sectionally pseudocompelemented lattices forms a variety of lattices which can be described by two simple identities. This variety has nice congruence properties. We summarize properties of sectionally pseudocomplemented posets and show differences to relative pseudocomplementation. We prove that every sectionally pseudocomplemented poset is completely L-semidistributive. We introduce the concept of congruence on these posets and show when the quotient structure becomes a poset again. Finally, we study the Dedekind-MacNeille completion of sectionally pseudocomplemented posets. We show that contrary to the case of relatively pseudocomplemented posets, this completion need not be sectionally pseudocomplemented but we present the construction of a so-called generalized ordinal sum which enables us to construct the Dedekind-MacNeille completion provided the completions of the summands are known.

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arXiv2019

Residuation in lattice effect algebras

We introduce the concept of a quasiresiduated lattice and prove that every lattice effect algebra can be organized into a commutative quasiresiduated lattice with divisibility. Also conversely, every such a lattice can be converted into a lattice effect algebra and every lattice effct algebra can be reconstructed form its assigned quasiresiduated lattice. We apply this method also for lattice pseudoeffect algebras introduced recently by Dvurecenskij and Vetterlein. We show that every good lattice pseudoeffect algebra can be organized into a (possibly non-commutative) quasiresiduated lattice with divisibility and conversely, every such a lattice can be converted into a lattice pseudoeffect algebra. Moreover, also a good lattice pseudoeffect algebra can be reconstructed from the assigned quasiresiduated lattice.

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arXiv2019

Relatively residuated lattices and posets

It is known that every relatively pseudocomplemented lattice is residuated and, moreover, it is distributive. Unfortunately, non-distributive lattices with a unary operation satisfying properties similar to relative pseudocomplementation cannot be converted in residuated ones. The aim of our paper is to introduce a more general concept of a relative residuated lattice in such a way that also non-modular sectionally pseudocomplemented lattices are included. We derive several properties of relative residuated lattices which are similar to those known for residuated ones and extend our results to posets.

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arXiv2019

Ideals of direct products of rings

It is known that an ideal of a direct product of commutative unitary rings is directly decomposable into ideals of the corresponding factors. We show that this does not hold in general for commutative rings and we find necessary and sufficient conditions for direct decomposability of ideals. For varieties of commutative rings we derive a Mal'cev type condition characterizing direct decomposability of ideals and we determine explicitly all varieties satisfying this condition.

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arXiv2019

Directly decomposable ideals and congruence kernels of commutative semirings

As pointed out in the monographs by J. S. Golan and by W. Kuich and A. Salomaa on semirings, ideals play an important role despite the fact that they need not be congruence kernels as in the case of rings. Hence, having two commutative semirings S1 and S2, one can ask whether an ideal I of their direct product S = S1 x S2 can be expressed in the form I1 x I2 where Ij is an ideal of Sj for j=1,2. Of course, the converse is elementary, namely if Ij is an ideal of Sj for j=1,2 then I1 x I2 is an ideal of S1 x S2. Having a congruence on a commutative semiring S, its 0-class is an ideal of S, but not every ideal is of this form. Hence, the lattice Id S of all ideals of S and the lattice Ker S of all congruence kernels (i.e. 0-classes of congruences) of S need not be equal. Furthermore, we show that the mapping which assigns to every congruence its kernel need not be a homomorphism from Con S onto Ker S. Moreover, the question arises when a congruence kernel of the direct product S1 x S2 of two commutative semirings can be expressed as a direct product of the corresponding kernels on the factors. In the paper we present necessary and sufficient conditions for such direct decompositions both for ideals and for congruence kernels of commutative semirings. We also provide sufficient conditions for varieties of commutative semirings to have directly decomposable kernels.

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arXiv2018

Residuation in modular lattices and posets

We show that every complemented modular lattice can be converted into a left residuated lattice where the binary operations of multiplication and residuum are term operations. The concept of an operator left residuated poset was introduced by the authors recently. We show that every strongly modular poset with complementation as well as every strictly modular poset with complementation can be organized into an operator left residuated poset in such a way that the corresponding operators M(x,y) and R(x,y) can be expressed by means of the operators L and U in posets. We describe connections between the operator left residuation in these posets and the residuation in their lattice completion. We also present examples of strongly modular and strictly modular posets.

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