MV-algebras: a variety for magnitudes with archimedean elements
by
Daniele Mundici
Dept. Computer Science, Univ. Milan, Via Comelico 39-41, 20135 Milan, Italy

Ralph McKenzie dealt with the question when two classes of algebras are equivalent as categories. He considered not only varieties and quasivarieties, but also more general classes. MV-algebras and lattice ordered abelian groups with strong unit yield an interesting example of a variety being categorically equivalent to a non-elementary class of algebras. A strong unit u in a lattice ordered abelian group G is an archimedean element, in the sense that elements of the form u, 2u, 3u, ... are cofinal in the positive cone of G. An MV-algebra is an abelian semigroup with neutral element 1 and an operation * satisfying the equations x** = x, x1* = 1* and (yx*)*y = (xy*)*x. MV-algebras were invented by C.C.Chang to prove the completeness theorem for the infinite-valued propositional calculus of Lukasiewicz. We shall discuss this categorical equivalence and its neighbours, linking (via Grothendieck's K_O functor) MV-algebras to the AF C*-algebras of infinite spin systems.

REFERENCES McKenzie, Ralph, An algebraic version of categorical equivalence for varieties and more general algebraic categories. Ursini, Aldo (ed.) et al., Logic and algebra. Proceedings of the international conference dedicated to the memory of Roberto Magari, April 26-30, 1994, Pontignano, Italy. New York, NY: Marcel Dekker. Lect. Notes Pure Appl. Math. 180, 211-243 (1996).

Cignoli, Roberto L.O.; D'Ottaviano, Itala M.L.; Mundici, Daniele, Algebraic foundations of many-valued reasoning. Trends in Logic-Studia Logica Library. 7. Dordrecht: Kluwer Academic Publishers. ix, 231 p.

Date received: January 3, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caig-49.