THE LINDENBAUM ALGEBRA OF THE THEORY OF THE CLASS OF ALL FINITE MODELS

Abstract

In this paper, we investigate the Lindenbaum algebra ℒ(T fin ) of the theory T fin = Th (M fin ) of the class M fin of all finite models of a finite rich signature. We prove that this algebra is an atomic Boolean algebra while its Gödel numeration γ is a [Formula: see text]-numeration. Moreover, the quotient algebra (ℒ(T fin )/ℱ, γ/ℱ) modulo the Fréchet ideal ℱ is a [Formula: see text]-algebra, which is universal over the class of all [Formula: see text] Boolean algebras. These conditions characterize uniquely the algebra ℒ(T fin ); moreover, these conditions characterize up to recursive isomorphism the numerated Boolean quotient algebra (ℒ(T fin )/ℱ, γ/ℱ). These results extend the work of Trakhtenbrot [17] and Vaught [18] on the first order theory of the class of all finite models of a finite rich signature.