## About some generalizations of ($$\lambda$$ ,$$\mu$$ )-compactness.(English)Zbl 0659.03020

The author presents without proof some interesting results concerning the compactness properties of model theoretic logics. He considers the following notion related to ($$\lambda$$,$$\mu)$$- and [$$\lambda$$,$$\mu$$ ]- compactness: A logic L is almost $$\kappa$$-($$\lambda$$,$$\mu)$$-compact, iff for every $$\Gamma$$ $$\subseteq L$$ of cardinality $$\leq \kappa$$, and for every $$\Sigma$$ $$\subseteq L$$ of cardinality $$\lambda$$, it holds: if for every subset $$\Sigma$$ ’ of $$\Sigma$$ of cardinality $$<\mu$$ there is a model of $$\Gamma$$ $$\cup \Sigma '$$, then for some $$\Sigma^*\subseteq \Sigma$$ of cardinality $$\lambda$$, $$\Gamma \cup \Sigma^*$$ has a model. $$\kappa$$-($$\lambda$$,$$\mu)$$-compactness is the same with the stronger requirement $$\Sigma^*=\Sigma$$. Among others, the author states the following results. (a) Every $$(\lambda^+,\lambda^+)$$-compact logic is almost $$\lambda^+$$-($$\lambda$$,$$\lambda)$$-compact. (b) If either $$\lambda$$ is regular or $$2^{\lambda}=\lambda^+$$, then every $$(\lambda^+,\lambda^+)$$-compact logic is ($$\lambda$$,$$\lambda)$$- compact. (c) Every $$2^{\lambda}$$-($$\lambda^+,\lambda^+)$$-compact logic is ($$\lambda$$,$$\lambda)$$-compact. The paper is connected to the work of J. A. Makowsky and S. Shelah [Ann. Pure Appl. Logic 25, 263-299 (1983; Zbl 0544.03013)] on $$[\lambda^+,\lambda^+]$$- compactness and regularity properties of ultrafilters.
Reviewer: J.Oikkonen

### MSC:

 03C95 Abstract model theory 03C20 Ultraproducts and related constructions 03E05 Other combinatorial set theory

Zbl 0544.03013