ACTA issues

Distributive lattices with sectionally antitone involutions

I. Chajda, R. Halaš, J. Kühr

Acta Sci. Math. (Szeged) 71:1-2(2005), 19-33

Abstract. We study upper-bounded lattices and join-semilattices that have antitone involutions in all intervals $[x,1]$ (sectionally antitone involutions). On such a (semi)lattice we introduce a binary operation the properties of which characterize the original order and sectionally antitone involutions. It turns out that bounded lattices with sectionally antitone involutions satisfying a simple additional condition are distributive and correspond one-to-one to well-known $MV$-algebras.

AMS Subject Classification (1991): MSC: 06D05, 06D35, 08B05

Keyword(s): Distributive lattice, MV, -algebra, sectionally antitone involution, chain

Received September 30, 2003, and in revised form October 4, 2004. (Registered under 5854/2009.)