Simplicial structures in MV-algebras and logic

Classical logic, as is well known, can be analyzed in a great part by algebraic methods using the Lindenbaum algebra obtained from the formal system. Since Chang [4, 5], Łukasiewicz logic has also been analyzed algebraically through the associated Lindenbaum type algebra. In this case this algebra is an MV-algebra [4]. Once again logical notions have an algebraic counterpart, for example, completeness relates strongly to semisimplicity [4, 5]. However, unlike the classical case where the algebras in question are Boolean and always semisimple, not all MV-algebras are semisimple. This fact, in a sense, enriches the theory of MV-algebras. Now every MV-algebra can be considered a Lindenbaum type algebra, namely an algebra associated to Łukasiewicz logic with additional axioms. Thus we can carry over to any MV-algebra various logical notions such as (in)completeness, consistency, satisfiability, etc. Two important logical notions are those of “formal consequence” and “semantical consequence”. Informally call these relations F, S respectively; consider them as binary relations. Now the completeness theorem just states F = S. Thus we can talk about an MV-algebra being “complete” provided the associated relations F, S are equal. In the case where the associated relations are not equal we shall compare them by using simplicial complexes. This provides a kind of “measure of the completeness” of a given MV-algebra. We attach to F, S respectively, simplicial complexes thereby enabling a comparison of F, S by comparing the simplicial structures. When F = S we say the algebra is “logically complete”; when the simplices are the same we say the algebra is “simplicially complete”, and when the homology groups are the same we say the algebra is “homologically complete”. We begin a study of MV-algebras from this point of view.