# zbMATH — the first resource for mathematics

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.
Reviewer: H.-J.Vogel

##### MSC:
 03C52 Properties of classes of models 08C10 Axiomatic model classes 03C60 Model-theoretic algebra
##### Keywords:
strong epimorphism; axiomatizable class