Characterization of axiomatizable classes with strong epimorphisms.

*(Russian)*Zbl 0711.03011
Tr. Inst. Mat. 12, 24-39 (1989).

In an earlier paper the author proved that in an axiomatizable class \({\mathcal K}\) all homomorphisms are strong if and only if any predicate P of \({\mathcal K}\) fulfils any so-called S-axiom. In the present paper the author gives a similar characterization of the strongness of epimorphisms between models. A homomorphism (epimorphism) \(\lambda\) from a model \({\mathfrak M}\) into a model \({\mathfrak N}\) is strong iff for all elements \(a_ 1,...,a_ n\in {\mathfrak M}\) and any predicate P such that \({\mathfrak N}\vDash P(\lambda (a_ 1),...,\lambda (a_ n))\) there exist \(b_ 1,...,b_ n\in {\mathfrak M}\) such that \({\mathfrak M}\vDash P(b_ 1,...,b_ n)\) and \(\lambda (a_ i)=\lambda (b_ i)\) (1\(\leq i\leq n)\). An S-axiom for P is an expression for P of a well-defined form in the given signature.

Using well-known basic definitions and results of model theory, given in papers of A. I. Mal’tsev, the author introduces the notion of SE-axiom similarly to the notion of S-axiom and shows that there are elements \(b_ 1,...,b_ n\in {\mathfrak M}\) for any elements \(a_ 1,...,a_ n\in {\mathfrak M}\) and any epimorphism \(\lambda\) such that \(\lambda (a_ i)=\lambda (b_ i)\) (1\(\leq i\leq n)\) and \({\mathfrak M}\vDash P(b_ 1,...,b_ n)\), if \({\mathfrak N}\vDash P(\lambda (a_ 1),...,\lambda (a_ n))\), \({\mathfrak M}\vDash \neg P(a_ 1,...,a_ n)\), and P fulfils an SE- axiom (Lemma 1).

Further, the author formulates four important properties for pairs of sentence sets of signature \(\Sigma '\) (\(\Sigma\cup \{c_ 0,...,c_ n\}\subseteq \Sigma '\subseteq \Sigma \cup \{c_ i|\) \(i<\kappa \})\), where \(\Sigma\) is a given signature, \(\kappa\) is an infinite cardinal with \(\kappa >| \Sigma |\), and \(\{c_ i|\) \(i<\kappa \}\) is a set of two by two different constant symbols which are not contained in \(\Sigma\).

Using these properties the author proves in detail that it is possible to find extensions \((X',Y')\) of a pair (X,Y) with the four important properties which fulfil the four conditions, too (Lemma 2-Lemma 6).

The main result is the following Theorem. In any axiomatizable class \({\mathcal K}\) all epimorphisms between models are strong if and only if every predicate fulfils an SE-axiom. - Necessity is given by Lemma 1. To show that the condition is sufficient the author assumes that in \({\mathcal K}\) all epimorphisms are strong and there is a predicate P which does not fulfil any SE-axiom. This assumption leads to a contradiction by using the results of the lemmata. A corollary is the following fact: Let \({\mathcal K}\) be an axiomatizable class of signature \(\Sigma\) such that all epimorphisms are strong and let \(\Sigma_ 0\) be any signature with \(\Sigma_ 0\subseteq \Sigma\). Then there exists an extension \(\Sigma '\) of \(\Sigma_ 0\) such that \(\Sigma_ 0\subseteq \Sigma '\subseteq \Sigma\), \(| \Sigma '| \leq \Sigma_ 0+\omega\), and in \({\mathcal K}\upharpoonright \Sigma '\) all epimorphisms are strong.

Using well-known basic definitions and results of model theory, given in papers of A. I. Mal’tsev, the author introduces the notion of SE-axiom similarly to the notion of S-axiom and shows that there are elements \(b_ 1,...,b_ n\in {\mathfrak M}\) for any elements \(a_ 1,...,a_ n\in {\mathfrak M}\) and any epimorphism \(\lambda\) such that \(\lambda (a_ i)=\lambda (b_ i)\) (1\(\leq i\leq n)\) and \({\mathfrak M}\vDash P(b_ 1,...,b_ n)\), if \({\mathfrak N}\vDash P(\lambda (a_ 1),...,\lambda (a_ n))\), \({\mathfrak M}\vDash \neg P(a_ 1,...,a_ n)\), and P fulfils an SE- axiom (Lemma 1).

Further, the author formulates four important properties for pairs of sentence sets of signature \(\Sigma '\) (\(\Sigma\cup \{c_ 0,...,c_ n\}\subseteq \Sigma '\subseteq \Sigma \cup \{c_ i|\) \(i<\kappa \})\), where \(\Sigma\) is a given signature, \(\kappa\) is an infinite cardinal with \(\kappa >| \Sigma |\), and \(\{c_ i|\) \(i<\kappa \}\) is a set of two by two different constant symbols which are not contained in \(\Sigma\).

Using these properties the author proves in detail that it is possible to find extensions \((X',Y')\) of a pair (X,Y) with the four important properties which fulfil the four conditions, too (Lemma 2-Lemma 6).

The main result is the following Theorem. In any axiomatizable class \({\mathcal K}\) all epimorphisms between models are strong if and only if every predicate fulfils an SE-axiom. - Necessity is given by Lemma 1. To show that the condition is sufficient the author assumes that in \({\mathcal K}\) all epimorphisms are strong and there is a predicate P which does not fulfil any SE-axiom. This assumption leads to a contradiction by using the results of the lemmata. A corollary is the following fact: Let \({\mathcal K}\) be an axiomatizable class of signature \(\Sigma\) such that all epimorphisms are strong and let \(\Sigma_ 0\) be any signature with \(\Sigma_ 0\subseteq \Sigma\). Then there exists an extension \(\Sigma '\) of \(\Sigma_ 0\) such that \(\Sigma_ 0\subseteq \Sigma '\subseteq \Sigma\), \(| \Sigma '| \leq \Sigma_ 0+\omega\), and in \({\mathcal K}\upharpoonright \Sigma '\) all epimorphisms are strong.

Reviewer: H.-J.Vogel