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Distributive lattices with sectionally antitone involutions. (English) Zbl 1099.06006
Suppose that each segment $$[x,1]$$ in a bounded distributive lattice $$L$$ is equipped with an order-reversing involution $$a\to a^x$$. Suppose the operation $$x\cdot y=(xvy)^y$$ satisfies the exchange condition $$x\cdot (y\cdot z)=y\cdot (x\cdot z)$$. Then the operations $$\neg x=x \cdot 0$$ and $$x\oplus y=(x\cdot 0)\cdot y$$ make $$L$$ into an MV-algebra $$M$$. Conversely, in any MV-algebra $$M$$, defining $$a^x=\neg a\oplus x$$ and $$x\cdot y=\neg x\oplus y$$ one obtains a bounded distributive lattice $$L$$ with order-reversing involution, and the induced operation $$\cdot$$ has the exchange property. This interesting way to obtain MV-algebras from enriched bounded distributive lattices is the main result of the paper. As is well known, MV-algebras are categorically equivalent lattice-ordered abelian groups with strong unit (see the present reviewer’s paper [J. Funct. Anal. 65, 15–63 (1986; Zbl 0597.46059)]). They were invented by Chang as the algebras of Łukasiewicz infinite-valued logic. For background on MV-algebras see the monograph [R. L. O. Cignoli, I. M. L. D’Ottaviano and D. Mundici, Algebraic foundations of many-valued reasoning. Dordrecht: Kluwer (2000; Zbl 0937.06009)].

##### MSC:
 06D05 Structure and representation theory of distributive lattices 06D35 MV-algebras