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Discussiones Mathematicae General Algebra and Applications 24(1) (2004) 31-42
DOI: https://doi.org/10.7151/dmgaa.1073

BOUNDED LATTICES WITH ANTITONE INVOLUTIONS AND PROPERTIES OF MV-ALGEBRAS

Ivan Chajda Department of Algebra and Geometry Palacký University, Faculty of Sciences Tomkova 40, 779-00 Olomouc Czech Republik e-mail: chajda@inf.upol.cz

Peter Emanovský Department of Mathematics Palacký University, Pedagogical Faculty Zizkovo nám. 5, 771-40 Olomouc Czech Republik e-mail:eman@pdfnw.upol.cz

Abstract

We introduce a bounded lattice L = (L;Ú,Ů,0,1), where for each p Î L there exists an antitone involution on the interval [p,1]. We show that there exists a binary operation · on L such that L is term equivalent to an algebra A(L) = (L;·,0) (the assigned algebra to L) and we characterize A(L) by simple axioms similar to that of Abbott's implication algebra. We define new operations Ĺ and Ř on A(L) which satisfy some of the axioms of MV-algebra. Finally we show what properties must be satisfied by L or A(L) to obtain all axioms of MV-algebra.

Keywords: antitone involution, distributive lattice, implication algebra, MV-algebra.

2000 Mathematics Subject Classification: 03C65, 03G25, 06A11, 06A12, 06D35, 08C10, 30N02.

References

[1]	J.C. Abbott, Semi-boolean algebra, Mat. Vestnik 4 (1967), 177-198.
[2]	R.L.O. Cignoli, I.M.L. D'Ottaviano and D. Mundici, Algebraic Foundations of Many-valued Reasoning, Kluwer Acad. Publ., Dordrecht/Boston/London 2000.
[3]	I. Chajda and R. Halas, Abbott's groupoids, Multiple Valued Logic, to appear.
[4]	I. Chajda, R. Halas and J. Kühr, Distributive lattices with sectionally antitone involutions, preprint 2003.

Received 20 June 2003
Revised 30 April 2004