IDEALS AND CONGRUENCES IN GENERALIZED MV-ALGEBRAS

Abstract

1. Properties of generalized MV-algebras MV-algebras, as algebras of type (©, O, 0,1) of signature (2,2,1,0,0), have been introduced and studied by C. C. Chang in [5] and [6] as an algebraic counterpart of the Lukasiewicz infinite valued propositional logic.Commutative lattice ordered groups (Z-groups) give a wide class of examples of MV-algebras.Let Q = (G, +, 0, -(•), V, A) be a commutative Z-group and 0 < u e G. Let us denote by [0, u] = {x e G;0 < x < u} the closed interval between 0 and u and put x © y = (x + y) A u,->x -u -x, and xQy = -i(-ix©-iy) for any x,y G [0,n].Then T(G,u) = ([0,u],©, ©,->,0,n) is an MV-algebra.D. Mundici proved in [10] that the class of such MValgebras is sufficiently general because every MV-algebra is isomorphic to r(G, u) for some commutative Z-group Q and some u € G, where u is a strong (order) unit in Q.Moreover, MV-algebras are categorically equivalent to commutative Z-groups with strong units (considered with unital homomorphisms).Commutative Z-groups are only very special cases of Z-groups because they form the smallest non-trivial variety of Z-groups, (see e.g.[7] or [9]).Hence the following non-commutative generalization of MV-algebras was introduced in [11].