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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 \((w \oslash u) \oslash v = (w \oslash v) \oslash u\).

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