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Quantum logic in dagger kernel categories
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"... This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial inject ..."
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This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical/ordertheoretic properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, orthomodularity, atomicity and completeness. For instance, the Sasaki hook and andthen connectives are obtained, as adjoints, via the existentialpullback adjunction between fibres. 1
Involutive categories and monoids, with a GNScorrespondence
 In Quantum Physics and Logic (QPL
, 2010
"... This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vecto ..."
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Cited by 4 (2 self)
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This paper develops the basics of the theory of involutive categories and shows that such categories provide the natural setting in which to describe involutive monoids. It is shown how categories of EilenbergMoore algebras of involutive monads are involutive, with conjugation for modules and vector spaces as special case. The core of the socalled GelfandNaimarkSegal (GNS) construction is identified as a bijective correspondence between states on involutive monoids and inner products. This correspondence exists in arbritrary involutive symmetric monoidal categories. 1
Convexity, duality, and effects
 IFIP Theoretical Computer Science 2010, number 82(1) in IFIP Adv. in Inf. and Comm. Techn
, 2010
"... This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call ‘convex functors’; they include what are usually called probability distribution functors. T ..."
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Cited by 3 (2 self)
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This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call ‘convex functors’; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are ‘dual ’ adjunctions for convex sets, one time with the Boolean truth values {0, 1} as dualising object, and one time with the probablity values [0, 1]. The third adjunction is between effect algebras and convex functors. 1
Partial algebras for ̷lukasiewicz logic and its extensions
, 2004
"... It is a wellknown fact that MValgebras, the algebraic counterpart of ̷Lukasiewicz logic, correspond to a certain type of partial algebras: latticeordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct ..."
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It is a wellknown fact that MValgebras, the algebraic counterpart of ̷Lukasiewicz logic, correspond to a certain type of partial algebras: latticeordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓgroups in a straightforward manner. In this paper, we consider several logics differing from ̷Lukasiewicz logics in that they contain further connectives: the P̷L, P̷L ′, P̷L ′ △, and ̷LΠlogics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing frings. All in all, we get threefold correspondences: the total algebras the partial algebras the representing rings. 1
Toward problems for mathematical fuzzy logic, in
 Proc. of IEEE International Conference on Fuzzy Systems
, 2006
"... The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1 ..."
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The paper discusses some open problems in the field of mathematical fuzzy logic which may have a decisive influence for the future development of fuzzy logic within the next decade. 1
Probabilities, Distribution Monads, and Convex Categories
"... Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again g ..."
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Probabilities are understood abstractly as forming a monoid in the category of effect algebras. They can be added, via a partial operation, and multiplied. This generalises key properties of the unit interval [0, 1]. Such effect monoids can be used to define a probability distribution monad, again generalising the situation for [0, 1]probabilities. It will be shown that there are translations backandforth, in the form of an adjunction, between effect monoids and “convex ” monads. This convexity property is formalised, both for monads and for categories. In the end this leads to “triangles of adjunctions ” (in the style of Coumans and Jacobs) relating all the three relevant structures: probabilities, monads, and categories. 1
IFSAEUSFLAT 2009 Poincaré recurrence theorem in MValgebras
"... Abstract — The classical Poincaré weak recurrence theorem states that for any probability space (Ω, S,P), any Pmeasure preserving transformation T, and any A ∈S, almost all points of A return to A. In the present paper the Poincaré theorem is proved when the σalgebra S is replaced by any σcomplet ..."
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Abstract — The classical Poincaré weak recurrence theorem states that for any probability space (Ω, S,P), any Pmeasure preserving transformation T, and any A ∈S, almost all points of A return to A. In the present paper the Poincaré theorem is proved when the σalgebra S is replaced by any σcomplete MValgebra.
MValgebra
"... MValgebras were introduced by Chang in 1958; algebraic basis for manyvalued logic. Equivalent definition by Mangani 1973: An MValgebra (M; ⊕, ∗, 0) is a (2, 1, 0) type of algebra: (MV1) the binary operation ⊕ is commutative and associative with the nullary operation 0 as neutral element; (MV2) a ..."
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MValgebras were introduced by Chang in 1958; algebraic basis for manyvalued logic. Equivalent definition by Mangani 1973: An MValgebra (M; ⊕, ∗, 0) is a (2, 1, 0) type of algebra: (MV1) the binary operation ⊕ is commutative and associative with the nullary operation 0 as neutral element; (MV2) a ⊕ 1 = 1 where 1 = 0 ∗; (MV3) 1 ∗ = 0; (MV4) (a ∗ ⊕ b) ∗ ⊕ b = (b ∗ ⊕ a) ∗ ⊕ a.MVpairs and states – p. 3/2 (MV4) is called Łukasiewicz axiom. It guarantees that MValgebra is a distributive lattice according to the ordering given by a ≤ b iff a ∗ ⊕ b = 1. Boolean algebras coincide with MValgebras satisfying the additional condition x ⊕ x = x. Mundici 1986: MValgebras are in categorical equivalence with [0, u] in Abelian latticeordered groups with strong unit u. A prototypical model of MValgebra is the real unit interval [0, 1].MVeffect algebras Chovanec and Kôpka: MValgebras are a subclass of a more general algebraic structures called effect algebras. An effect algebra (Foulis and Bennett 1994; Kôpka and
Coreflections in Algebraic Quantum Logic ∗
"... Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular posets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a comb ..."
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Various generalizations of Boolean algebras are being studied in algebraic quantum logic, including orthomodular lattices, orthomodular posets, orthoalgebras and effect algebras. This paper contains a systematic study of the structure in and between categories of such algebras. It does so via a combination of totalization (of partially defined operations) and transfer of structure via coreflections. 1
Symmetry, Integrability and Geometry: Methods and Applications Archimedean Atomic Lattice Effect Algebras with Complete Lattice of Sharp Elements ⋆
"... doi:10.3842/SIGMA.2010.001 Abstract. We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra E is sepa ..."
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doi:10.3842/SIGMA.2010.001 Abstract. We study Archimedean atomic lattice effect algebras whose set of sharp elements is a complete lattice. We show properties of centers, compatibility centers and central atoms of such lattice effect algebras. Moreover, we prove that if such effect algebra E is separable and modular then there exists a faithful state on E. Further, if an atomic lattice effect algebra is densely embeddable into a complete lattice effect algebra Ê and the compatiblity center of E is not a Boolean algebra then there exists an (o)continuous subadditive state on E. Key words: effect algebra; state; sharp element; center; compatibility center 2010 Mathematics Subject Classification: 06C15; 03G12; 81P10 1 Introduction, basic definitions and some known facts The classical (Kolmogorovian) probability theory is assuming that every two events are simultaniously measurable (as every two elements of a Boolean algebra are mutually compatible). Thus this theory cannot explain events occuring, e.g., in quantum physics, as well as in economy and many other areas.