Logical and algebraic properties of generalized orthomodular posets (original) (raw)
Related papers
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...
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.
Variety of orthomodular posets
Miskolc Mathematical Notes
Orthomodular posets play an important role in the so-called logical structure of a physical system as formerly pointed out by numerous authors. In particular, they play an essential role in the logic of quantum mechanics. To avoid usual problems with partial algebras, we define the so-called orthomodular directoid as an everywhere defined algebra and we show that every orthomodular poset can be converted into an orthomodular directoid and vice versa. Since orthomodular directoids are defined equationally, they form a variety having nice congruence properties.
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.
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.
Residuated Operators and Dedekind–MacNeille Completion
arXiv: Logic, 2018
The concept of operator residuation for bounded posets with unary operation was introduced by the first two authors. It turns out that in some cases when these operators are transformed into lattice terms and the poset mathbfP{\mathbf P}mathbfP is completed into a Dedekind-MacNeille completion BDM(mathbfP)\BDM(\mathbf P)BDM(mathbfP) then the complete lattice BDM(mathbfP)\BDM(\mathbf P)BDM(mathbfP) becomes a residuated lattice with respect to these transformed terms. It is shown that this holds in particular for Boolean posets and for relatively pseudocomplemented posets. More complicated situation is with orthomodular and pseudo-orthomodular posets. We show which operators MMM (multiplication) and RRR (residuation) yield operator left-residuation in a pseudo-orthomodular poset mathbfP{\mathbf P}mathbfP and if BDM(mathbfP)\BDM(\mathbf P)BDM(mathbfP) is an orthomodular lattice then the transformed lattice terms odot\odotodot and to\toto form a left residuation in BDM(mathbfP)\BDM(\mathbf P)BDM(mathbfP). However, it is a problem to determine when BDM(mathbfP)\BDM(\mathbf P)BDM(mathbfP) is an orthomodular lattice. We get some...
On Some New Operations on Orthomodular Lattices
International Journal of Theoretical Physics, 2000
Kotas conditionals are used to define six pairs of disjunction- andconjunction-like operations on orthomodular lattices. Although five of them necessarily differfrom the lattice operations on elements that are not compatible, they coincidewith the lattice operations on all compatible elements of the lattice and theydefine on the underlying set a partial order relation that coincides with the originalone. Some of the new operations are noncommutative on noncompatible elements,but this does not exclude the possibility to endow them with a physicalinterpretation. The new operations are in general nonassociative, but for someof them a Foulis—Holland-type theorem concerning associativity instead ofdistributivity holds. The obtained results suggest that these new operations canserve as alternative algebraic models for the logical operations of disjunctionand conjunction.
Notre Dame Journal of Formal Logic, 2000
New conjunctionlike and disjunctionlike operations on orthomodular lattices are defined with the aid of formal Mackey decompositions of not necessarily compatible elements. Various properties of these operations are studied. It is shown that the new operations coincide with the lattice operations of join and meet on compatible elements of a lattice but they necessarily differ from the latter on all elements that are not compatible. Nevertheless, they define on an underlying set the partial order relation that coincides with the original one. The new operations are in general nonassociative: if they are associative, a lattice is necessarily Boolean. However, they satisfy the Foulis-Holland-type theorem concerning associativity instead of distributivity.
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.