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prof. RNDr. Pavel Pták, DrSc.

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A Symmetric-Difference-Closed Orthomodular Lattice That Is Stateless

  • Authors: Voráček, V., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS. 2023, 40(2), 397-402. ISSN 0167-8094.
  • Year: 2023
  • DOI: 10.1007/s11083-022-09621-7
  • Link: https://doi.org/10.1007/s11083-022-09621-7
  • Department: Department of Mathematics
  • Annotation:
    This paper carries on the investigation of the orthomodular lattices that are endowed with a symmetric difference. Let us call them ODLs. Note that the ODLs may have a certain bearing on "quantum logics" - the ODLs are close to Boolean algebras though they capture the phenomenon of non-compatibility. The initial question in studying the state space of the ODLs is whether the state space can be poor. This question is of a purely combinatorial nature. In this note, we exhibit a finite ODL whose state space is empty (respectively, whose state space is a singleton).

On Blocks in the Products and Ultraproducts of Orthomodular Lattices

  • Authors: Matousek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2023, 62(11), ISSN 0020-7748.
  • Year: 2023
  • DOI: 10.1007/s10773-023-05488-5
  • Link: https://doi.org/10.1007/s10773-023-05488-5
  • Department: Department of Mathematics
  • Annotation:
    Let OML denote the class of orthomodular lattices (OMLs, quantum logics). Let L be an OML and let B be a maximal Boolean subalgebra of L. Then B is called a block of L. In the algebraic investigation of OMLs a natural question is whether the blocks of a product (resp. ultraproduct) of OMLs are products (resp. ultraproducts) of the blocks of the respective "coordinate" OMLs. We first add to the study of this question as regards the products and the centres of the products (a special mention deserves the result that the centre of the ultraproduct is the ultraproduct of the centres of the respective OMLs). Then we pass to the analogous questions for ultraproducts where we present main results of this note. Though this question on the "regular" behaviour of blocks in ultraproducts remains open in general, we provide a positive partial solution. This contributes to the understanding of varieties important to quantum theories - to the varieties that contain both set-representable OMLs and projection OMLs. We consider an axiomatizable class of the OMLs, OMLn, whose blocks uniformly intersect in finite sets of the maximal cardinality of 2(n). It is worth realizing within the connection to quantum logic theory that, for instance, the OMLs given by Greechie diagrams belong to OML2. The importance of the results is commented on in relation to the state space properties of OMLs.

On locally finite orthomodular lattices

  • DOI: 10.1515/ms-2023-0040
  • Link: https://doi.org/10.1515/ms-2023-0040
  • Department: Department of Mathematics
  • Annotation:
    Let us denote by LF the class of all orthomodular lattices (OMLs) that are locally finite (i.e., L ∈ LF provided each finite subset of L generates in L a finite subOML). In this note, we first show how one can obtain new locally finite OMLs from the initial ones and enlarge thus the class LF. We find LF considerably large though, obviously, not all OMLs belong to LF. Then we study states on the OMLs of LF. We show that local finiteness may to a certain extent make up for distributivity. For instance, we show that if L ∈ LF and if for any finite subOML K there is a state s: K → [0,1] on K, then there is a state on the entire L. We also consider further algebraic and state properties of LF relevant to the quantum logic theory.

On the set-representable orthomodular posets that are point-distinguishing

  • Authors: Burešová, D., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2023, 62 ISSN 0020-7748.
  • Year: 2023
  • DOI: 10.1007/s10773-023-05436-3
  • Link: https://doi.org/10.1007/s10773-023-05436-3
  • Department: Department of Mathematics
  • Annotation:
    Let us denote by SOMP the class of all set-representable orthomodular posets and by PDSOMP those elements of SOMP in which any pair of points in the underlying set P can be distinguished by a set (i.e., (P,L) ∈ PDSOMP precisely when for any pair x,y ∈ P there is a set A ∈ L with x ∈ A and y ∈/ A). In this note we first construct, for each (P, L) ∈ SOMP, a point-distinguishing orthomodular poset that is isomorphic to (P,L). We show that by using a generalized form of the Stone representation technique we also obtain point-distinguishing representations of ( P , L). We then prove that this technique gives us point-distinguishing representations on which all two-valued states are determined by points (all two-valued states are Dirac states). Since orthomodular posets may be regarded as abstract counterparts of event structures about quantum experiments, results of this work may have some relevance for the foundation of quantum mechanics.

Orthosystems of submodules of a module

  • DOI: 10.1080/00927872.2022.2164008
  • Link: https://doi.org/10.1080/00927872.2022.2164008
  • Department: Department of Mathematics
  • Annotation:
    Let M be a module over a ring. We first introduce a certain algebraic sub-system Sigma of the lattice of all submodules of M (an orthosystem of submod-ules). We then show that any ortholattice can be represented as a Sigma for a suitable module. Next, we introduce linear (resp. pre-Hilbert) ortholattices as those ortholattices that allow for a "linear" representation sigma (resp. for a meet-preserving "linear" representation sigma). These notions involve a type of splitting property of sigma. As an important example, we show that any Boolean algebra is pre-Hilbertian. We then find that linear orthosystems are orthomodular and that they satisfy the ortho-Arguesian law. In the rest, we consider complete orthosystems. We show that each complete orthosystem can be induced by an orthogonality relation &updatedExpOTTOM; on M. If (M, &updatedExpOTTOM;) is a linear orthospace on M then the collection of all &updatedExpOTTOM;-closed submodules is a complete orthosystem, and vice versa. Finally, we address a natural model theoretic question on the axiomati-zation of orthomodular orthosystems.& mdash;The results obtained may contribute to the algebraic foundation of quantum theory.

A Note on Extensions of Non-additive Measures

  • Authors: prof. RNDr. Pavel Pták, DrSc., Weber, H.
  • Publication: International Journal of Theoretical Physics. 2021, 60(2), 512-514. ISSN 0020-7748.
  • Year: 2021
  • DOI: 10.1007/s10773-019-04049-z
  • Link: https://doi.org/10.1007/s10773-019-04049-z
  • Department: Department of Mathematics
  • Annotation:
    The motivation for our consideration comes from the fuzzy set theory in a potential relation to quantum theories and mathematical economics: Given a certain non-additive assignment on a Boolean algebra (a kind of "belief measure"), can this assignment be extended over a larger Boolean algebra? We answer this question in the affirmative. By examining the universality of the method used, we conclude that even when we let the assignment subject to an arbitrary collection of unsharp inequalities, we are always able to extend the measure so defined over a larger Boolean algebra.

On Frink Ideals in Orthomodular Posets

  • DOI: 10.1007/s11083-020-09537-0
  • Link: https://doi.org/10.1007/s11083-020-09537-0
  • Department: Department of Mathematics, Machine Learning
  • Annotation:
    Let S denote the class of orthomodular posets in which all maximal Frink ideals are selective. Let R (resp. T) be the class of orthomodular posets defined by the validity of the following implications: P is an element of R if the implication a, b is an element of P, a boolean AND b = 0 double right arrow a <= b' holds (resp., P is an element of T if the implication a. b = a boolean AND b' = 0 double right arrow a = 0 holds). In this note we prove the following slightly surprising result: R subset of S subset of T. Since orthomodular posets are often understood as quantum logics, the result might have certain bearing on quantum axiomatics.

Quantum logics defined by divisibility conditions

  • DOI: 10.1007/s10773-018-3977-y
  • Link: https://doi.org/10.1007/s10773-018-3977-y
  • Department: Department of Mathematics, Machine Learning
  • Annotation:
    Let p be a prime number and let S be a countable set. Let us consider the collection DivSp of all subsets of S whose cardinalities are multiples of p and the complements of such sets. Then the collection DivSp constitutes a (set-representable) quantum logic (i.e., DivSp is an orthomodular poset). We show in this note that each state on DivSp can be extended over the Boolean algebra exp S of all subsets of S.

Quantum Logics that are Symmetric-difference-closed

  • Authors: Burešová, D., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2021, 60(10), 3919-3926. ISSN 0020-7748.
  • Year: 2021
  • DOI: 10.1007/s10773-021-04950-6
  • Link: https://doi.org/10.1007/s10773-021-04950-6
  • Department: Department of Mathematics
  • Annotation:
    In this note we contribute to the recently developing study of "almost Boolean" quantum logics (i.e. to the study of orthomodular partially ordered sets that are naturally endowed with a symmetric difference). We call them enriched quantum logics (EQLs). We first consider set-representable EQLs. We disprove a natural conjecture on compatibility in EQLs. Then we discuss the possibility of extending states and prove an extension result for Z(2)-states on EQLs. In the second part we pass to general orthoposets with a symmetric difference (GEQLs). We show that a simplex can be a state space of a GEQL that has an arbitrarily high degree of noncompatibility. Finally, we find an appropriate definition of a "parametrization" as a mapping between GEQLs that preserves the set-representation.

Jauch-Piron states on quantum logics

  • Authors: Hroch, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Journal of Algebra and Its Applications (JAA). 2020, 19(1), 1-5. ISSN 0219-4988.
  • Year: 2020
  • DOI: 10.1142/S0219498820500176
  • Link: https://doi.org/10.1142/S0219498820500176
  • Department: Department of Mathematics
  • Annotation:
    We show in this note that if B is a Boolean subalgebra of the lattice quantum logic L, then each state on B can be extended over L as a Jauch-Piron state provided L is Jauch-Piron unital with respect to B (i.e. for each nonzero b is an element of B, there is a Jauch-Piron state s on L such that s(b) = 1). We then discuss this result for the case of L being the Hilbert space logic L(H) and L being a set-representable logic.

Orthomodular lattices that are Z(2)-rich

  • DOI: 10.1007/s11587-018-0378-8
  • Link: https://doi.org/10.1007/s11587-018-0378-8
  • Department: Department of Mathematics
  • Annotation:
    We study the orthomodular lattices (OMLs) that have an abundance of Z(2)-valued states. We call these OMLs Z(2)-rich. Themotivation for the investigation comes from a natural algebraic curiosity that reflects the state of the (orthomodular) art, the consideration also has a certain bearing on the foundation of quantum theories (OMLs are often identified with " quantum logics") and mathematical logic (Z(2)-states are fundamental in mathematical logic). Before we launch on the subject proper, we observe, for a potential application elsewhere, that there can be a more economic introduction of Z(2)-richness - the Z(2)-richness in the orthocomplemented setup is sufficient to imply orthomodularity. In the further part we review basic examples of OMLs that are Z(2)-rich and that are not. Then we show, as a main result, that the Z(2)-rich OMLs form a large and algebraicly "friendly" class-they form a variety. In the appendix we note that the OMLs that allow for a natural introduction of a symmetric difference provide a source of another type of examples of Z(2)-rich OMLs. We also formulate open questions related to the matter studied.

A NOTE ON FIELD-VALUED MEASURES

  • Authors: De Simone, A., Hroch, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Mathematica Slovaca. 2017, 67(6), 1295-1300. ISSN 0139-9918.
  • Year: 2017
  • DOI: 10.1515/ms-2017-0052
  • Link: https://doi.org/10.1515/ms-2017-0052
  • Department: Department of Mathematics
  • Annotation:
    We consider the Horn-Tarski condition for the extension of (signed) measures (resp., non-negative measures) in the setup of field-valued assignments. For a finite collection C of subsets of Omega, we find that the extension from C over the collection exp Omega of all subsets of Omega is implied by, and indeed equivalent to, a certain type of Frobenius theorem (resp. a certain type of Farkas lemma). This links classical notions of linear algebra with a generalized version of Horn-Tarski condition on extensions of measures. We also observe that for a general (infinite) C the Horn-Tarski condition guarantees the extension of signed measures (here the standard Zorn lemma applies). However, we find out that the extensions for non-negative ordered-field-valued measures are generally not available. (C) 2017 Mathematical Institute Slovak Academy of Sciences

Concrete Quantum Logics and Delta-Logics, States and Delta-States

  • Authors: Hroch, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2017, 56(12), 3852-3859. ISSN 0020-7748.
  • Year: 2017
  • DOI: 10.1007/s10773-017-3359-x
  • Link: https://doi.org/10.1007/s10773-017-3359-x
  • Department: Department of Mathematics
  • Annotation:
    By a concrete quantum logic (in short, by a logic) we mean the orthomodular poset that is set-representable. If L = (Omega, L) is a logic and L is closed under the formation of symmetric difference, Delta, we call L a Delta-logic. In the first part we situate the known results on logics and states to the context of Delta-logics and Delta-states (the Delta-states are the states that are subadditive with respect to the symmetric difference). Moreover, we observe that the rather prominent logic epsilon(even)(Omega) of all even- coeven subsets of the countable set Omega possesses only Delta-states. Then we show when a state on the logics given by the divisibility relation allows for an extension as a state. In the next paragraph we consider the so called density logic and its Delta-closure. We find that the Delta-closure coincides with the power set. Then we investigate other properties of the density logic and its factor.

Varieties of Orthocomplemented Lattices Induced by Lukasiewicz-Groupoid-Valued Mappings

  • Authors: Matousek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2017, 56(12), 4004-4016. ISSN 0020-7748.
  • Year: 2017
  • DOI: 10.1007/s10773-017-3411-x
  • Link: https://doi.org/10.1007/s10773-017-3411-x
  • Department: Department of Mathematics
  • Annotation:
    In the logico-algebraic approach to the foundation of quantum mechanics we sometimes identify the set of events of the quantum experiment with an orthomodular lattice ("quantum logic"). The states are then usually associated with (normalized) finitely additive measures ("states"). The conditions imposed on states then define classes of orthomodular lattices that are sometimes found to be universal-algebraic varieties. In this paper we adopt a conceptually different approach, we relax orthomodular to orthocomplemented and we replace the states with certain subadditive mappings that range in the Aukasiewicz groupoid. We then show that when we require a type of "fulness" of these mappings, we obtain varieties of orthocomplemented lattices. Some of these varieties contain the projection lattice in a Hilbert space so there is a link to quantum logic theories. Besides, on the purely algebraic side, we present a characterization of orthomodular lattices among the orthocomplemented ones. - The intention of our approach is twofold. First, we recover some of the Mayet varieties in a principally different way (indeed, we also obtain many other new varieties). Second, by introducing an interplay of the lattice, measure-theoretic and fuzzy-set notions we intend to add to the concepts of quantum axiomatics.

States On Orthocomplemented Difference Posets (Extensions)

  • Authors: Hroch, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Letters in Mathematical Physics. 2016, 106(8), 1131-1137. ISSN 0377-9017.
  • Year: 2016
  • DOI: 10.1007/s11005-016-0862-6
  • Link: https://doi.org/10.1007/s11005-016-0862-6
  • Department: Department of Mathematics
  • Annotation:
    We continue the investigation of orthocomplemented posets that are endowed with a symmetric difference (ODPs). The ODPs are orthomodular and, therefore, can be viewed as "enriched" quantum logics. In this note, we introduced states on ODPs. We derive their basic properties and study the possibility of extending them over larger ODPs. We show that there are extensions of states from Boolean algebras over unital ODPs. Since unital ODPs do not, in general, have to be set-representable, this result can be applied to a rather large class of ODPs. We then ask the same question after replacing Boolean algebras with "nearly Boolean" ODPs (the pseudocomplemented ODPs). Making use of a few results on ODPs, some known and some new, we construct a pseudocomplemented ODP, P, and a state on P that does not allow for extensions over larger ODPs.

STATES WITH VALUES IN THE LUKASIEWICZ GROUPOID

  • DOI: 10.1515/ms-2015-0139
  • Link: https://doi.org/10.1515/ms-2015-0139
  • Department: Department of Mathematics
  • Annotation:
    In this paper we consider certain groupoid-valued measures and their connections with quantum logic states. Let * stand for the Lukasiewicz t-norm on [0, 1](2). Let us consider the operation lozenge on [0, 1] by setting x lozenge y = (x(perpendicular to)*y(perpendicular to))(perpendicular to) *(x*y)(perpendicular to), where x(perpendicular to) = 1-x. Let us call the triple L = ([0, 1], lozenge, 1) the Lukasiewicz groupoid. Let B be a Boolean algebra. Denote by L(B) the set of all L-valued measures (L-valued states). We show as a main result of this paper that the family L(B) consists precisely of the union of classical real states and Z(2)-valued states. With the help of this result we characterize the L-valued states on orthomodular posets. Since the orthomodular posets are often understood as "quantum logics" in the logico-algebraic foundation of quantum mechanics, our approach based on a fuzzy-logic notion actually select a special class of quantum states. As a matter of separate interest, we construct an orthomodular poset without any L-valued state. (C) 2016 Mathematical Institute Slovak Academy of Sciences

Characterization of Boolean Algebras in Terms of Certain States of Jauch-Piron Type

  • Authors: Matoušek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2015, 54(12), 4476-4481. ISSN 0020-7748.
  • Year: 2015
  • DOI: 10.1007/s10773-015-2638-7
  • Link: https://doi.org/10.1007/s10773-015-2638-7
  • Department: Department of Mathematics
  • Annotation:
    Suppose that L is an orthomodular lattice (a quantum logic). We show that L is Boolean exactly if L possesses a strongly unital set of weakly Jauch-Piron states, or if L possesses a unital set of weakly positive states. We also discuss some general properties of Jauch-Piron-like states.

On the Farkas lemma and the Horn Tarski measure-extension theorem

  • Authors: De Simone, A., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Linear Algebra and Its Applications. 2015, 481 243-248. ISSN 0024-3795.
  • Year: 2015
  • DOI: 10.1016/j.laa.2015.05.002
  • Link: https://doi.org/10.1016/j.laa.2015.05.002
  • Department: Department of Mathematics
  • Annotation:
    We first derive a certain version of the Farkas lemma called the 0-1 Farkas lemma (the 0-1 FL). We then show that the 0-1 FL is equivalent to a measure-extension theorem. By applying one implication of this result, we prove that the 0-1 FL implies the classical Horn-Tarski measure-extension theorem.

States on systems of sets that are closed under symmetric difference

  • DOI: 10.1002/mana.201500029
  • Link: https://doi.org/10.1002/mana.201500029
  • Department: Department of Mathematics, Department of Cybernetics
  • Annotation:
    We consider extensions of certain states. The states are defined on the systems of sets that are closed under the formation of the symmetric difference (concrete quantum logics). These systems can be viewed as certain set-representable quantum logics enriched with the symmetric difference. We first show how the compactness argument allows us to extend states on Boolean algebras over such systems of sets. We then observe that the extensions are sometimes possible even for non-Boolean situations. On the other hand, a difference-closed system can be constructed such that even two-valued states do not allow for extensions.

Orthomodular Posets Related to Z2-Valued States

  • Authors: Matoušek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2014, 53(10), 3323-3332. ISSN 0020-7748.
  • Year: 2014
  • DOI: 10.1007/s10773-013-1690-4
  • Link: https://doi.org/10.1007/s10773-013-1690-4
  • Department: Department of Mathematics
  • Annotation:
    We study orthocomplemented posets (certain quantum logics) that possess an abundance of Z 2-valued states. We first discuss their basic properties and, by means of examples, we illuminate intrinsic qualities of these orthocomplemented posets. We then address the problem of axiomatizability of our class of posets—a question that appears natural from the algebraic point of view. In the last section we show, as a main result, that supports of the posets endowed with symmetric difference constitute an important example of orthocomplemented posets under consideration. This result is obtained by a thorough analysis of certain types of ideals.

Orthocomplemented difference lattices in association with generalized rings.

  • DOI: 10.2478/s12175-012-0064-3
  • Link: https://doi.org/10.2478/s12175-012-0064-3
  • Department: Department of Mathematics
  • Annotation:
    Orthocomplemented difference lattices (ODLs) are orthocomplemented lattices endowed with an additional operation of "abstract symmetric difference". In studying ODLs as universal algebras or instances of quantum logics, several results have been obtained (see the references at the end of this paper where the explicite link with orthomodularity is discussed, too). Since the ODLs are "nearly Boolean", a natural question arises whether there are "nearly Boolean rings" associated with ODLs. In this paper we find such an association - we introduce some difference ring-like algebras (the DRAs) that allow for a natural one-to-one correspondence with the ODLs. The DRAs are defined by only a few rather plausible axioms. The axioms guarantee, among others, that a DRA is a group and that the association with ODLs agrees, for the subrings of DRAs, with the famous Stone (Boolean ring) correspondence.

ORTHOCOMPLEMENTED DIFFERENCE LATTICES WITH FEW GENERATORS

  • Department: Department of Mathematics
  • Annotation:
    The algebraic theory of quantum logics overlaps in places with certain areas of cybernetics, notably with the field of artificial intelligence (see, e.g., [19, 20]). Recently an effort has been exercised to advance with logics that possess a symmetric difference ([13, 14]) - with so called orthocomplemented difference lattices (ODLs). This paper further contributes to this effort. In [13] the author constructs an ODL that is not set-representable. This example is quite elaborate. A main result of this paper somewhat economizes on this construction: There is an ODL with 3 generators that is not set-representable (and so the free ODL with 3 generators cannot be set-representable). The result is based on a specific technique of embedding orthomodular lattices into ODLs. The ODLs with 2 generators are always set-representable as we show by characterizing the free ODL with 2 generators - this ODL is MO(3) x 2(4).

Measures on circle coarse-grained systems of sets

  • DOI: 10.1007/s11117-009-0015-6
  • Link: https://doi.org/10.1007/s11117-009-0015-6
  • Department: Department of Mathematics
  • Annotation:
    We show that a (non-negative) measure on a circle coarse-grained system of sets can be extended, as a (non-negative) measure, over the collection of all subsets of the circle. This result contributes to quantum logic probability (de Lucia in Colloq Math 80(1):147-154, 1999; Gudder in Quantum Probability, Academic Press, San Diego, 1988; Gudder in SIAM Rev 26(1):71-89, 1984; Harding in Int J Theor Phys 43(10):2149-2168, 2004; Navara and Ptak in J Pure Appl Algebra 60:105-111, 1989; Ptak in Proc Am Math Soc 126(7):2039-2046, 1998, etc.) and completes the analysis of coarse-grained measures carried on in De Simone and Ptak (Bull Pol Acad Sci Math 54(1):1-11, 2006; Czechoslov Math J 57(132) n.2:737-746, 2007), Gudder and Marchand (Bull Pol Acad Sci Math 28(11-12):557-564, 1980) and Ovchinnikov (Construct Theory Funct Funct Anal 8:95-98, 1992).

On identities in orthocomplemented difference lattices

  • DOI: 10.2478/s12175-010-0033-7
  • Link: https://doi.org/10.2478/s12175-010-0033-7
  • Department: Department of Mathematics
  • Annotation:
    In this note we continue the investigation of algebraic properties of orthocomplemented (symmetric) difference lattices (ODLs) as initiated and previously studied by the authors. We take up a few identities that we came across in the previous considerations. We first see that some of them characterize, in a somewhat non-trivial manner, the ODLs that are Boolean. In the second part we select an identity peculiar for set-representable ODLs. This identity allows us to present another construction of an ODL that is not set-representable. We then give the construction a more general form and consider algebraic properties of the 'orthomodular support'.

Orthocomplemented Posets with a Symmetric Difference

  • Authors: Matoušek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: ORDER-A JOURNAL ON THE THEORY OF ORDERED SETS AND ITS APPLICATIONS. 2009, 26(1), 1-21. ISSN 0167-8094.
  • Year: 2009

Symmetric difference on orthomodular lattices and $Z_2$-valued states

  • Authors: Matoušek, M., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Commentationes Mathematicae Universitatis Carolinae. 2009, 50(4), 535-547. ISSN 0010-2628.
  • Year: 2009
  • Department: Department of Mathematics
  • Annotation:
    The investigation of orthocomplemented lattices with a symmetric difference initiated the following question: Which orthomodular lattice can be embedded in an orthomodular lattice that allows for a symmetric difference? In this paper we present a necessary condition for such an embedding to exist. The condition is expressed in terms of $Z_2$-valued states and enables one, as a consequence, to clarify the situation in the important case of the lattice of projections in a Hilbert space.

Extending States on Finite Concrete Logics (vol 44, 2005)

  • DOI: 10.1007/s10773-006-9298-6
  • Link: https://doi.org/10.1007/s10773-006-9298-6
  • Department: Department of Mathematics, Department of Cybernetics
  • Annotation:
    We summarize and extend results about ``small'' quantum structures with small dimensions of state spaces. These constructions have contributed to the theory of orthomodular lattices. More general quantum structures (orthomodular posets, orthoalgebras, and effect algebras) admit sometimes simplifications, but there are problems where no progress has been achieved.

Group-valued Measures on Coarse-Grained Quantum Logics

  • Authors: de Simone, A., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Czechoslovak Mathematical Journal. 2007, 2007(57), 737-746. ISSN 0011-4642.
  • Year: 2007
  • DOI: 10.1007/s10587-007-0110-4
  • Link: https://doi.org/10.1007/s10587-007-0110-4
  • Department: Department of Mathematics
  • Annotation:
    In [3] it was shown that a (real) signed measure on a cyclic coarse-grained quantum logic can be extended, as a signed measure, over the entire power algebra. Later ([9]) this result was re-proved (and further improved on) and, moreover, the non-negative measures were shown to allow for extensions as non-negative measures. In both cases the proof technique used was the technique of linear algebra. In this paper we further generalize the results cited by extending group-valued measures on cyclic coarse-grained quantum logics (or non-negative group-valued measures for lattice-ordered groups). Obviously, the proof technique is entirely different from that of the preceding papers. In addition, we provide a new,combinatorial argument for describing all atoms of cyclic coarse-grained quantum logics.

Quantum Logics as Underlying Structures of Generalized Probability Theory

  • Authors: prof. RNDr. Pavel Pták, DrSc., Pulmannova, S.
  • Publication: Handbook of Quantum Logic and Quantum Structures. Amsterdam: Elsevier, 2007. p. 147-213. ISBN 978-0-444-52870-4.
  • Year: 2007

Extending Coarse-Grained Measures

  • Authors: De Simone, A., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Bulletin of the Polish Academy of Sciences. 2006, 54(1), 1-11. ISSN 0239-7285.
  • Year: 2006
  • Department: Department of Mathematics
  • Annotation:
    The paper deals with the extension problem of measures on the coarse=grained quantum logics.

Extending States on Finite Concrete Logics

  • DOI: 10.1007/s10773-005-7083-6
  • Link: https://doi.org/10.1007/s10773-005-7083-6
  • Department: Department of Mathematics, Department of Cybernetics
  • Annotation:
    In this note we collect several observations on state extensions. They may be instrumental to anyone who pursues the theory of quantum logics. In particular, we find out when extensions (resp. signed extensions) exist in the "concrete" concrete logic of all even-element subsets of an even-element set. We also mildly add to the study of difference-closed logics by finding an extension theorem for subadditive states.

On the (Non)existence of States on Orthogonally Closed Subspaces in an Inner Product Space

  • Authors: prof. RNDr. Pavel Pták, DrSc., Svozil, K., Chetcuti, E.
  • Publication: International Journal of Theoretical Physics. 2005, 44(7), 1023-1028. ISSN 0020-7748.
  • Year: 2005

Relatively Additive States on Quantum Logics

  • Authors: prof. RNDr. Pavel Pták, DrSc., Weber, H.
  • Publication: Commentationes Mathematicae Universitatis Carolinae. 2005, 46(2), 327-338. ISSN 0010-2628.
  • Year: 2005
  • Department: Department of Mathematics
  • Annotation:
    In the paper, a variant of partially additive states---the states which are additive with respect to a given Boolean subalgebra---on quantum logics (that is, orthomodular posets) is investigated. Examples of quantum logics which possess, or do not possess, different kinds of partially additive states are constructed. In the constructions, rather advanced orthomodular combinatorics is used.

Cantor-Bernstein theorems for quantum structures

For n>=5 there is no nontrivial Z_2-measure on L(R^n)

On prehilbert-space logics

  • Authors: prof. RNDr. Pavel Pták, DrSc., Chetcuti, E.
  • Publication: Proceedings of the 3rd International Workshop on Nonstandard Logics. Praha: ČVUT v Praze, FEL, 2004. pp. 26-31. ISBN 80-01-03003-2.
  • Year: 2004
  • Department: Department of Mathematics
  • Annotation:
    The paper provides a simple proof of a completeness characterization of prehilbert spaces and discusses certain properties related to non-standard (quantum) mathematical logics.

Order Properties of Splitting Subspaces in an Inner Product Space

  • Department: Department of Mathematics
  • Annotation:
    The main result of this paper asserts that the orthomodular poset of all splitting subspaces of an inner product space does not have to possess the Riesz Interpolation Property.

Quantum Logics with the Riesz Interpolation Property

  • Authors: prof. RNDr. Pavel Pták, DrSc., Dvurečenskij, A.
  • Publication: Mathematische Nachrichten. 2004, 2004(271), 10-14. ISSN 0025-584X.
  • Year: 2004

Regular measures on tribes of fuzzy sets

Convex structure of the space of fuzzy measures

On States on Orthogonally Closed Subspaces of an Inner Product Space

The Vitali-Hahn-Saks theorem for the product of quantum logics

Convex Structure of the Space of T-Measures

Lattice Properties of Subspace Families in an Inner Product Space

  • Authors: prof. RNDr. Pavel Pták, DrSc., Weber, H.
  • Publication: Proceedings of the American Mathematical Society. 2001, 129(7), 2111-2117. ISSN 0002-9939.
  • Year: 2001

On Interval Homogeneous Orthomodular Lattices

On Interval Homogeneous Orthomodular Lattices

On the Set Representation of an Orthomodular Poset

Cantor-Bernstein Theorems for Noncommutative Structures

Concrete Quantum Logics

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 2000, 39(3), 827-838. ISSN 0020-7748.
  • Year: 2000

Mathematical Models of Uncertainty

Nearness in Digital Images and Proximity Spaces

  • Authors: prof. RNDr. Pavel Pták, DrSc., Kropatsch, W.
  • Publication: Discrete Geometry for Computer Imagery. Berlin: Springer, 2000. pp. 69-77. ISBN 3-540-41396-0.
  • Year: 2000

Observables in the Logico-Algebraic Approach

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Current Research in Operational Quantum Logic. Dordrecht: Kluwer Academic Publishers, 2000. p. 67-80. ISBN 0-7923-6258-6.
  • Year: 2000

On the de Morgan Property of the Standard Brouwer-Zadeh Poset

Orthomodular Lattices with State-Separated Noncompatibles Pairs

  • Authors: Mayet, R., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Czechoslovak Mathematical Journal. 2000, 50(125), 359-366. ISSN 0011-4642.
  • Year: 2000

Quantum Logics with Classically Determined States

Quasivarieties of Orthomodular Lattices Determined by Conditions on States

Uncertainty and dependence in classical and quantum logic - the role of triangular norms

Concrete logics

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Abstracts Quantum Structures '98. Liptovský Mikuláš: Vojenská akadémia, 1998, pp. 71.
  • Year: 1998

Considering Uncertainty and Dependence in Boolean, Quantum and Fuzzy Logics

Quantum Logics with Given Centers and Variable State Spaces

Some Nearly Boolean Orthomodular Posets

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Proceedings of the American Mathematical Society. 1998, 126(7), 2039-2046. ISSN 0002-9939.
  • Year: 1998
  • DOI: 10.1090/S0002-9939-98-04403-7
  • Link: https://doi.org/10.1090/S0002-9939-98-04403-7
  • Department: Department of Mathematics
  • Annotation:
    Let L be an orthomodular partially ordered set ("a quantum logic"). Let us say that L is nearly Boolean if L is set-representable and if every state on L is subadditive. We first discuss conditions under which a nearly Boolean OMP must be Boolean. Then we show that in general a nearly Boolean OMP does not have to be Boolean. Moreover, we prove that an arbitrary Boolean algebra may serve as the centre of a (non-Boolean) nearly Boolean OMP.

The path-connectedness in Z2 and Z3 and classical topologies (the point-neighbourhood formalism)

  • Authors: Kropatsch, W., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Advances in Pattern Recognition, Proceedings of Joint IAPR International Workshop SSPR'98 and SPR'98. Berlin: Springer, 1998. pp. 181-189. ISBN 3-540-64858-5.
  • Year: 1998

Types of Uncertainty and the Role of the Frank t-Norms in Classical and Nonclassical Logics

Difference Posets and Orthoalgebras

Digital Topologies Revisited: An Approach Based on the Topological Point Neighbou

  • Authors: prof. RNDr. Pavel Pták, DrSc., Kofler, H., Kropatsch, W.
  • Publication: Discrete Geometry for Computer Imagery. Berlin: Springer, 1997. pp. 151-159. ISBN 3-540-63884-9.
  • Year: 1997

Introduction to Linear Algebra

Observables

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Proceedings Summer School on Measure Theory and Real Analysis. Trieste: University Trieste, 1997. pp. 124-136.
  • Year: 1997

Two Remarks on Inner Product Spaces

  • Authors: prof. RNDr. Pavel Pták, DrSc., Weber, H.
  • Publication: Contributions to General Algebra 10. Preceedings of the Klagenfurt Conference. Klagenfurt: Verlag Joh. Heyn, 1997. pp. 123-135. ISBN 3-85366-890-9.
  • Year: 1997

Types of Uncertainly Types of Dependence

Coming from Distributive to Orthomodular

The Dog-and-Rabbit Chase Revisited

  • Department: Department of Mathematics
  • Annotation:
    For the dog-and-rabbit-chase problem we find an explicit description of the dog's path and the duration of the chase.

Almost Boolean Algebras as Quantum Logics

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Abstract 10th International Congress of Logic, Methodology and Philosophy of Science. Florence: ???, 1995. pp. 438.
  • Year: 1995

On absolutely compatible elements and hidden variables in quantum logics

On the Tensor Product of a Boolen Algebra and an Orthoalgebra

  • Authors: Foulis, D., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Czechoslovak Mathematical Journal. 1995, 45(1), 117-126. ISSN 0011-4642.
  • Year: 1995

State Space of an Orthoalgebra

States on orthoalgebras

A Measure-Theoretic Characterization of Boolean Algebras Amony Orthomodular Lattices

  • Authors: prof. RNDr. Pavel Pták, DrSc., Pulmannová, S.
  • Publication: Commentationes Mathematicae Universitatis Carolinae. 1994, 35(1), 205-208. ISSN 0010-2628.
  • Year: 1994

A Note on Inner Product Spaces

Mathematical Methods of Quantum Theories

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: CTU Seminar 94. Praha: České vysoké učení technické v Praze, 1994. pp. 43-44.
  • Year: 1994

States on Orthostructures (Noncommutative Measure Theory)

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Rendiconti dellmInstituto di Mathematica dellmUniversitá di Trieste. Trieste: LINT, 1994. pp. 37-49.
  • Year: 1994

Central Envelopes of Orthomodular Lattices

Enveloping Classical and Quantum Logics (Macroscoping Measuring Apparatures and Quantum Events)

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Symposium on the Foundations of Modern Physics. Singapore: World Scientific, 1993. pp. 338-340. ISBN 981-02-1507-X.
  • Year: 1993

Jauch-Piron Property (Everywhere!) in the Logico-Algebraic Foundations of Quantum Theories

  • Authors: prof. RNDr. Pavel Pták, DrSc.,
  • Publication: International Journal of Theoretical Physics. 1993, 32(3), 1985-1990. ISSN 0020-7748.
  • Year: 1993

Orthomodular Structures as Quantum Logics

P-measures on Soft Fuzzy Sigma-algebras

Concrete Quantum Logics With Covering Properties

Hilbert-space-valued States on Guantum Logics

States on soft fuzzy algebras |

Hilbert-space-valued Measures on Boolean Algebras (Extensions)

  • Department: Department of Mathematics
  • Annotation:
    We prove that if B1 is a Boolean subalgebra of B2 and if m: B1 ! H is a bounded finitely additive measure, where H is a Hilbert space, then m admits an extension over B2. This result generalizes the well-known result for real-valued measures (see e.g. [1]). Then we consider orthogonal measures as a generalization of two-valued measures. We show that the latter result remains valid for dimH < 1. If dimH = 1, we are only able to prove a weaker result: If B1 is a Boolean subalgebra of B2 and m: B1 ! H is an orthogonal measure, then we can find a Hilbert space K such that H K and such that there is an orthogonal measure k : B2 ! K with k/B1 = m.

Orthomodular structures as quantum logics

  • Department: Department of Mathematics
  • Annotation:
    The quantum logic approach to quantum mechanics is a foundational investigation based on the mathematical structures of orthomodular lattices or posets.

Almost Boolean orthomodular posets

Enlargements of logics (sigma-orthocomplete case)

ENLARGEMENTS OF QUANTUM-LOGICS

A completeness Criterion for Inner Product-Spaces

Quantum logics with Jauch-Piron states

Measures on orthomodular partially ordered sets

  • Authors: prof. RNDr. Pavel Pták, DrSc., Rogalewicz, V.
  • Publication: Journal of Pure and Applied Algebra. 1983, 28 75-80. ISSN 0022-4049.
  • Year: 1983

Regularly full logics and the uniqueness problem for observables

  • Authors: Rogalewicz, V., prof. RNDr. Pavel Pták, DrSc.,
  • Publication: Annales de l'Institut Henri Poincaré, Physique théorique. 1983, 38(1), 69-74. ISSN 0246-0211.
  • Year: 1983

Two-valued measures on sigma-classes

Responsible person Ing. Mgr. Radovan Suk