Some theorems on structural entailment relations

Abstract

The classesMatr($\subseteq$) of all matrices (models) for structural finitistic entailments$\subseteq$ are investigated. The purpose of the paper is to prove three theorems: Theorem I.7, being the counterpart of the main theorem from Czelakowski [3], and Theorems II.2 and III.2 being the entailment counterparts of Bloom's results [1]. Theorem I.7 states that if a classK of matrices is adequate for$\subseteq$, thenMatr($\subseteq$) is the least class of matrices containingK and closed under the formation of ultraproducts, submatrices, strict homomorphisms and strict homomorphic pre-images. Theorem II.2 in Section II gives sufficient and necessary conditions for a structural entailment to be finitistic. Section III contains theorems which characterize finitely based entailments.