Variety of orthomodular posets (original) (raw)
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Logical and algebraic properties of generalized orthomodular posets
arXiv: Logic, 2020
Generalized orthomodular posets were introduced recently by D. Fazio, A. Ledda and the first author of the present paper in order to establish a useful tool for studying the logic of quantum mechanics. They investigated structural properties of these posets. In the present paper we study logical and algebraic properties of these posets. In particular, we investigate conditions under which they can be converted into operator residuated structures. Further, we study their representation by means of algebras (directoids) with everywhere defined operations. We prove congruence properties for the class of algebras assigned to generalized orthomodular posets and, in particular, for a subvariety of this class determined by a simple identity. Finally, in contrast to the fact that the Dedekind-MacNeille completion of an orthomodular poset need not be an orthomodular lattice we show that the Dedekind-MacNeille completion of a stronger version of a generalized orthomodular poset is nearly an o...
The logic of orthomodular posets of finite height
Logic Journal of the IGPL
Orthomodular posets form an algebraic formalization of the logic of quantum mechanics. A central question is how to introduce implication in such a logic. We give a positive answer whenever the orthomodular poset in question is of finite height. The crucial advantage of our solution 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 that its implication operator is everywhere defined. We present here a 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.
Varieties corresponding to classes of complemented posets
arXiv: Rings and Algebras, 2019
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.
Orthomodular and generalized orthomodular posets
arXiv (Cornell University), 2022
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.
Studia Logica, 1995
The notion of unsharp orthoalgebra is introduced and it is proved that the category of unsharp orthoalgebras is isomorphic to the category of D-posers. A completeness theorem for some partial logics based on unsharp orthoalgebras, orthoalgebras and orthomodular posets is proved.
Orthomodular Lattices and a Quantum Algebra
International Journal of Theoretical Physics
We show that one can formulate an algebra with lattice ordering so as to contain one quantum and five classical operations as opposed to the standard formulation of the Hilbert space subspace algebra. The standard orthomodular lattice is embeddable into the algebra. To obtain this result we devised algorithms and computer programs for obtaining expressions of all quantum and classical operations within an orthomodular lattice in terms of each other, many of which are presented in the paper. For quantum disjunction and conjunction we prove their associativity in an orthomodular lattice for any triple in which one of the elements commutes with the other two and their distributivity for any triple in which a particular element commutes with the other two. We also prove that the distributivity of symmetric identity holds in Hilbert space, although whether or not it holds in all orthomodular lattices remains an open problem, as it does not fail in any of over 50 million Greechie diagrams...
Algebraic properties of paraorthomodular posets
2020
Paraorthomodular posets are bounded partially ordered set with an antitone involution induced by quantum structures arising from the logico-algebraic approach to quantum mechanics. The aim of the present work is starting a systematic inquiry into paraorthomodular posets theory both from an algebraic and order-theoretic perspective. On the one hand, we show that paraorthomodular posets are amenable of an algebraic treatment by means of a smooth representation in terms of bounded directoids with antitone involution. On the other, we investigate their order-theoretical features in terms of forbidden configurations. Moreover, sufficient and necessary conditions characterizing bounded posets with an antitone involution whose Dedekind-MacNeille completion is paraorthomodular are provided.
How to introduce the connective implication in orthomodular posets
Asian-European Journal of Mathematics
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.
On the Structure of Orthomodular Posets
It is shown how all orthomodular posets (of various kinds) are constructible from families of sets satisfying various conditions, usually with the generating family emerging as identical with (or contained in) the family of frames (that is, maximal orthogonal subsets of the non-zero elements) of the constructed orthomodular poset.
Almost Boolean orthomodular posets
Journal of Pure and Applied Algebra, 1989
Let 8 be the class of concrete (= set-representable) orthomodular partially ordered sets. Let I, be the class of Boolean OMP's (Boolean algebras). In-between g0 and 8 (+ZoC f?) there are three classes originating in quantum axiomatics -the class gt of concrete Jauch-Piron OMP's (gP E B, * if s(A) =s(B) = 1 for a state s on .YZ and A, B ed, then s(C) = 1 for some C l ._& with CCA flB), the class 9, of 'compact-like' OMP's (de g2 Q Sp is concrete and for every pair A,BE&! we have a finite d-covering of A fIB), and the class B, of 'infimum faithful' OMP's (.&E gs * if ar\b=O for a, bed then as b'). We study these classes and show that H?,c B, c 67, c B, c I. We also exhibit examples establishing that at least three of the latter inclusions are proper. Then we prove a representation theorem -every OMP is an epimorphic image of an OMP from K7,. Finally, we comment on the interpretation of the results in quantum axiomatics and formulate open questions.