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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)
##### Keywords:
difference poset; $$D$$-lattice; MV-algebra
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##### References:
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