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)].


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