# zbMATH — the first resource for mathematics

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)
##### Keywords:
difference poset; $$D$$-lattice; MV-algebra
Full Text:
##### References:
 [1] Chang, C. C. (1959). Algebraic analysis of many valued logics,Transactions of the American Mathematical Society,88, 467-490. · Zbl 0084.00704 [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 [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 [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 [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 [10] Sikorski, R. (1964).Boolean Algebras, Springer-Verlag, Berlin. · Zbl 0123.01303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.