Abstract

In Andreka-Nemeti [1] the class ST r of all small trees over C is dened for an arbitrary category C. Throughout the present paper C de- notes an arbitrary category. In Def. 4 of [1] on p. 367 the injectivity relation j= ) is dened. Intuitively the members of ST r represent the formulas and j= represents the validity relation be- tween objects of C considered as models and small trees of C considered as formulas. If ' 2 ST r and a 2 Ob C then the statement a j= ' is associated to the model theoretic statement \the formula ' is valid in the model a". It is proved there that the Los lemma is true in every category C if we use the above quoted concepts. To this the notion of an ultra- product Pi2Iai=U of objects 2 IOb C of C if we use the above quoted concepts. To this the notion of an ultraproduct Pi2Iai=U of objects 2 IOb C of C is dened in [1], in [2] and in [7] Def. 12. Then the problem was asked there \for which categories is the characterization theorem of axiomatizable hulls of classes of models Mod T h K = Uf U p K true?", where the operators Uf and U p on classes of models is dened on p. 319 of the book [3]. Of course, here in the denition of Uf and U p on classes K Ob C of objects of C we have to replace the standard notion of ultraproducts of models by the above quoted category theoretic ultraproduct Pi2Iai=U of objects of C. That is for any C and K Ob C we dene U p K exactly as in [2] p. 136, but a more detailed denition is found in [7] Def. 17