Extensions of the $\aleph_0$-valued Ł ukasiewicz propositional logic.
M. G. Beavers · Notre Dame Journal of Formal Logic · 1993
MV-algebras were introduced by Chang in 1958 preliminary to his providing an algebraic completeness proof for the K 0 -vaΓued Lukasiewicz propositional logic, L Xo .In this paper a method is given for determining, for an arbitrary normal extension of L Ko , an MV-algebra characteristic for the extension.The characteristic algebras are finite direct products of two sets of linearly ordered MV-algebras identified by Komori.Conversely, it is shown that, given an algebra which is isomorphic to the direct product of elements from these two sets of linearly ordered MV-algebras, a single axiom can be determined which, when added to the axioms for L Xo , yields an axiomatization sound and weakly complete for the given algebra.As a consequence of the lattice ordering of these products of MV-algebras the cardinal and ordinal degrees of completeness of any normal extension of L Ko can be determined.