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D-lattices. (English) Zbl 0840.03046

A \(D\)-poset [F. Chovanec and F. Kôpka, Math. Slovaca 44, 21-34 (1994; Zbl 0789.03048)] is a bounded poset \(P\) with a partial binary operation \(\backslash\) such that \(b \backslash a\) is defined iff \(a \leq b\), and for \(a,b,c \in P\), we have (i) if \(a \leq b\), then \(b \backslash a \leq b\) and \(b \backslash (b \backslash a) = a\); (ii) if \(a \leq b \leq c\), then \(c \backslash b \leq c \backslash a\) and \((c \backslash a) \backslash (c \backslash b) = b \backslash a\). If \(P\) is a lattice, then \(P\) is said to be a \(D\)-lattice. Then we can extend the partial operation \(\backslash\) to a (total) binary operation \(\oslash\) via \(b \oslash a = b \backslash (a \wedge b)\). The main result of the paper under review says that a \(D\)-lattice \(P\) can be organized into an MV-algebra iff, for all \(u,v,w \in P\), we have \((w \oslash u) \oslash v = (w \oslash v) \oslash u\).

MSC:

03G12 Quantum logic
06B99 Lattices
03B50 Many-valued logic
06D30 De Morgan algebras, Łukasiewicz algebras (lattice-theoretic aspects)

Citations:

Zbl 0789.03048
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Full Text: DOI

References:

[1] Chang, C. C. (1959). Algebraic analysis of many valued logics,Transactions of the American Mathematical Society,88, 467-490. · Zbl 0084.00704 · doi:10.1090/S0002-9947-1958-0094302-9
[2] Chovanec, F. (1993). States and observables on MV algebras,Tatra Mountains Mathematical Publications,3, 55-64. · Zbl 0799.03074
[3] Dvure?enskij, A., and Pulmannov?, S. (1994). Difference posets, effects, and quantum measurements,International Journal of Theoretical Physics,33, 819-850. · Zbl 0806.03040 · doi:10.1007/BF00672820
[4] Foulis, D. J., Greechie, R. J., and Ruttimann, G. T. (1992). Filters and supports in orthoalgebras,International Journal of Theoretical Physics,31, 789-807. · Zbl 0764.03026 · doi:10.1007/BF00678545
[5] K?pka, F. (1992). D-posets of fuzzy sets,Tatra Mountains Mathematical Publications,1, 83-88.
[6] K?pka, F., and Chovanec, F. (1994). D-posets,Mathematica Slovaca,44, 21-34. · Zbl 0789.03048
[7] Mundici, D. (1986). Interpretation of AFC *-algebras in Lukasiewicz sentential calculus,Journal of Functional Analysis,65, 15-63. · Zbl 0597.46059 · doi:10.1016/0022-1236(86)90015-7
[8] Pt?k, P., and Pulmannov?, S. (1991).Orthomodular Structures as Quantum Logics, VEDA, Bratislava, and Kluwer, Dordrecht. · Zbl 0743.03039
[9] Rie?anov?, Z., and Br?el, D. (1994). Contraexamples in difference posets and orthoalgebras,International Journal of Theoretical Physics,33, 133-141. · Zbl 0795.03092 · doi:10.1007/BF00671618
[10] Sikorski, R. (1964).Boolean Algebras, Springer-Verlag, Berlin. · Zbl 0123.01303
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