Stability of Regional Orthomodular Posets Under Synchronisation and Refinement

Abstract 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.

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DISJUNCTIVITY AND ORTHODISJUNCTIVITY IN ORTHOMODULAR POSETS

Demonstratio Mathematica ◽

10.1515/dema-1979-0416 ◽

1979 ◽

Vol 12 (4) ◽

Cited By ~ 1

Author(s):

Radoslaw Godowski

Keyword(s):

Orthomodular Posets

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On Orthomodular Posets Generated by Transition Systems

Electronic Notes in Theoretical Computer Science ◽

10.1016/j.entcs.2011.01.013 ◽

2011 ◽

Vol 270 (1) ◽

pp. 147-154 ◽

Cited By ~ 1

Author(s):

Luca Bernardinello ◽

Lucia Pomello ◽

Stefania Rombolà

Keyword(s):

Transition Systems ◽

Orthomodular Posets

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Weakly orthomodular and dually weakly orthomodular posets

Asian-European Journal of Mathematics ◽

10.1142/s1793557118500936 ◽

2018 ◽

Vol 11 (02) ◽

pp. 1850093 ◽

Cited By ~ 2

Author(s):

Ivan Chajda ◽

Helmut Länger

Keyword(s):

Quantum Mechanics ◽

Orthomodular Poset ◽

Algebraic Semantic ◽

Orthomodular Posets ◽

Congruence Properties

Orthomodular posets form an algebraic semantic for the logic of quantum mechanics. We show several methods how to construct orthomodular posets via a representation within the powerset of a given set. Further, we generalize this concept to the concept of weakly orthomodular and dually weakly orthomodular posets where the complementation need not be antitone or an involution. We show several interesting examples of such posets and prove which intervals of these posets are weakly orthomodular or dually weakly orthomodular again. To every (dually) weakly orthomodular poset can be assigned an algebra with total operations, a so-called (dually) weakly orthomodular [Formula: see text]-lattice. We study properties of these [Formula: see text]-lattices and show that the variety of these [Formula: see text]-lattices has nice congruence properties.

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