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.
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.
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
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.
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.
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.
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
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
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)
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.
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
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
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
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.
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.
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
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  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).
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
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
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)
We summarize and extend results about ``small'' quantum structures
with small dimensions of state
spaces. These constructions have contributed to the theory of
lattices. More general quantum structures (orthomodular posets,
and effect algebras) admit sometimes simplifications, but there are
where no progress has been achieved.
Group-valued Measures on Coarse-Grained Quantum Logics
In  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 () 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
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
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.
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)
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. ). 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.