Exotic logics

Pták, Pavel

doi:10.4064/cm-54-1-1-7

Cited by 20 publications

(15 citation statements)

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“…We first examine finite logics. Obviously, the problem degenerates if S(L) is very "poor" (which, unfortunately, may actually happen; see [2,5]). Following the usual terminology (see [3] …”

Section: Resultsmentioning

confidence: 99%

Quantum Logics with Lattice State Spaces

Binder¹,

Navara²

1987

Proceedings of the American Mathematical Society

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ABSTRACT. Let L be a quantum logic and let S(L) denote the set of all states on L. (By a state we mean a nonnegative bounded cr-additive measure, not necessarily normalized.)We ask whether every logic whose state space is a lattice has to be Boolean. We prove that this is so for finite logics and "projection logics." On the other hand, we show that there exist even concrete non-Boolean logics with a lattice state space (in fact, we prove that every countable concrete logic can be enlarged to a logic with a lattice state space). In the appendix we shortly consider the lattice properties of the set of observables and correct the paper [10].1. Introduction and preliminaries.In axiomatic quantum mechanics (see [3,8]), the "event structure" of a physical system is modelled by a cr-orthomodular partially ordered set (usually called a quantum logic). The states of the system then may be assumed to correspond to nonnegative bounded cr-additive measures on L, called "states" on L. Let us denote by S(L) the set of all states on L. If L is a Boolean cr-algebra, which is the case of an experiment subject to "classical mechanics," then S(L) is a lattice. In this paper we speculate whether this property of S(L) characterizes exactly "classical experiments."We find an axiomatic setup of the problem and obtain the results indicated in the abstract.Let us first review basic notions as we shall use them in the sequel. DEFINITION 1. A (quantum) logic is a set L equipped with a partial ordering < and a unary operation ', such that the following conditions are satisfied (the symbols A, V stand for the lattice operations induced by <):

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“…We first examine finite logics. Obviously, the problem degenerates if S(L) is very "poor" (which, unfortunately, may actually happen; see [2,5]). Following the usual terminology (see [3] …”

Section: Resultsmentioning

confidence: 99%

Quantum Logics with Lattice State Spaces

Binder¹,

Navara²

1987

Proceedings of the American Mathematical Society

View full text Add to dashboard Cite

show abstract

“…It remains to be shown that L possesses a state. Defineŝ(k, 1) = 1 for any k ∈ K andŝ(k, 0) = 0 for any k ∈ K. It can be easily seen thatŝ admits a unique extension to a state on K (see also [14]). This completes the proof.…”

Section: Basic Notionsmentioning

confidence: 99%