${\rm C}_1$ is not algebraizable.
Renato A. Lewin, Irene F. Mikenberg, M. G. Schwarze · Notre Dame Journal of Formal Logic · 1991
The purpose of this brief note is to give a very short proof of a fact, first proved by Mortensen, that illustrates the strength of the theory of algebraizability of deductive systems developed by Blok and Pigozzi.We build a small matrix model of C\ for which the Leibniz operator is not 1-1.In [3], da Costa introduces a family of paraconsistent deductive systems C n , n = 1,2,3,... among other things he raises the question of algebraizability of these systems.In [4] Mortensen proves that the paraconsistent deductive systems C n are not algebraizable by showing that in the absolutely free formula algebra for C x there is no nontrivial congruence compatible with the set of theorems of the system.Since Mortensen gives no formal definition of algebraizability and a related theory of algebraizability, his proof is long and somewhat complicated.Blok and Pigozzi develop such a theory in [1] using a generalization of the usual Lindenbaum-Tarski algebraization process.The reader is referred to that paper, especially Chapter 5, for the justification of our proof.The idea is very simple, it is proven (see Theorem 5.1) that if a deductive system is algebraizable and d = is a matrix model, then there is an isomorphism between the lattice of filters of d and the lattice of congruences of Q.Moreover, this isomorphism is given by the function which assigns to each filter Fthe largest compatible congruence, that is, the largest congruence θ such that if aθb and a E F, then b Gf.Thus, in order to prove nonalgebraizability of a system, it is enough to find a matrix model in which this function, called the Leibniz operator, is not an isomorphism.In Blok and Pigozzi [2], the question of the existence of a small matrix model for the system C i9 in which the Leibniz operator is not an isomorphism, is posed.We answer this question affirmatively, thus proving the nonalgebraizability of Cγ.Obviously, since the system C x is an extension of the systems C ni n = 1,2,3,..., none of these are algebraizable.