*3.7. Ultraproducts in Stratified Institutions*

The method of ultraproducts is renowed as extremely powerful and pervading a lot of deep results in model theory [2,88]. For instance, model ultraproducts are instrumental in the non-standard analysis [4,5] as the hyperreals are constructed by this technique. Chief among the ultraproduct method concepts and results that have been lifted to abstract institution theory is a very general version of Ło´s theorem obtained as a puzzle of preservation results [34,44]. Then, general compactness results have been obtained as a consequence of this. Furthermore, in [61], all these have been extended to the framework of modalised institutions. In [52], we took another step by generalising the developments of [61] to arbitrary stratified institutions. In what follows, we present the milestones of this development:

• For any filter *<sup>F</sup>* over a set *<sup>I</sup>* and for any family (*Mi*)*i*∈*<sup>I</sup>* of <sup>Σ</sup> models, its *F-product* is defined categorically as the co-limit *μ* of a diagram of projections:

where for each *<sup>J</sup>* ∈ *<sup>F</sup>*, (*pJ*,*<sup>j</sup>* : *MJ* → *Mj*)*j*∈*<sup>J</sup>* denotes a categorical product. This categorical approach on *filtered products* (called *ultraproducts* when *F* is an ultrafilter) has been used in various other categorical approaches to model theory such as [11,12,15,89], etc.

• The preservation of (the satisfaction of) a sentence *ρ* by *F*-filtered products is defined as follows. For any Σ sentence *ρ*, we introduce the following notation:

$$A\_{\mu}(\rho) = \bigcup\_{J \in F} [\![\mu\_{J}]\!] \bigcap\_{j \in J} [\![p\_{J,j}]\!]^{-1} [\![M\_{j}, \rho]\!].$$

Let F be a class of filters. Then, *ρ* is


for all filters *<sup>F</sup>* ∈ F and all families of models (*Mi*)*i*∈*I*. When the *<sup>F</sup>*-products are *concrete*, which means that they are preserved by the stratification—a very common situation in the applications—the stratified concept of preservation in S reduces to the ordinary institution theoretic concept of preservation in <sup>S</sup> .

• Then, we have developed a series of results expressing the invariance of preservation, corresponding to various connectives. In the case of the propositional connectives, this

invariance can be reduced to the corresponding invariance in ordinary institutions, which are already established in [34,44]. In the case of the quantifiers, this cannot be completed, but the proofs are similar to those from the ordinary institution theoretic framework. More interesting are the invariance results for modalities and nominals, as they do not have a counterpart in ordinary institutions, with the presence of stratification playing a key role. However, this is hardly unexpected, since the connectives are relevant only when models have internal states.


With respect to the compactness consequences of these invariances of preservation results, which together give a Ło´s-style theorem for abstract stratified institutions, both in the local and global flattening (i.e., <sup>S</sup> and <sup>S</sup>∗, respectively), we usually obtain the model compactness property. However, the entailment–theoretic compactness of the semantic consequence may be obtained only for <sup>S</sup> , as in <sup>S</sup><sup>∗</sup> negation, disjunction, existential quantifiers, etc., usually connectives that are related to negation in one form or another, pose some problems.
