*4.6. Preservation and Consequences*

In [95], there is a development of a body of preservation results in the same style as had been conducted for ordinary institutions in [34] or for stratified institutions in [52]. The milestones of this development are as follows:

	- **–** *e* is *κ-preserved by* F*-products* when for each *F*-product (*μ<sup>J</sup>* : *MJ* → *MF*)*J*∈*<sup>F</sup>* (where *F* ∈ F is a filter over *I*)

$$\{i \in I \mid (M\_i \mid = e) \not\supset \kappa\} \in F \text{ implies } (M\_F \mid = e) \not\supset \kappa;$$

**–** *e* is *κ-preserved by* F*-factors* when for each *F*-product as above we have the reverse implication to the above.

As a matter of terminology, when F is the class of all ultrafilters, we rather say directly "*κ*-preserved by ultraproducts/ultrafactors". When F is the class of all singleton filters, we rather say "*κ*-preserved by direct products/factors". In addition, when we do not specify the truth value *κ* and we just say "preserved by F-products/factors", we mean that the sentence is *κ*-preserved for *all* truth values *κ*.

Note that whilst *κ*-preservation represents just a rephrasing of the preservation concepts from binary institution theory because "*ρ* is *κ*-preserved by ..." is technically the same with "(*ρ*, *κ*) is preserved by ..." in the binary flattening, this is not the case for the preservation for *all* truth values. In other words "*ρ* is preserved by ..." in an L-institution cannot be reduced to preservation in its binary flattening of a single sentence.

	- **–** Invariance of preservation by F-products under ∧ and quantifications;
	- **–** Invariance of preservation by F-factors under ∧, ∨, ∗ and quantifications; and
	- **–** *ρ* ⇒ *ρ* is preserved by F-products when *ρ* is preserved by F-factors and *ρ* is preserved by F-products.

Each of these results is subject to some specific conditions of various intensities of a general nature regarding L, model reducts, F, etc. All of them are manageable in concrete applications.


In [95], two main consequences of these preservation results have been derived.


The former result involves also preservation by 'sub-models', which is a concept that is taken care of by the *inclusion systems* of [44,65], etc. (Such involvement of inclusion systems is common to all institution–theoretic approaches to quasi-varieties ([44]).)

For all this general theory, FMA presents itself as a special case when some general results cannot always be applied due to a lack of basic sentences. However, in [95], it is shown how an invariance of preservation results can still be used to obtain the preservation by filtered products for a relevant class of FMA sentences and consequently a model compactness result for those.

#### *4.7. Around Graded Interpolation*

In [109], the author developed a study of interpolation in the graded consequence framework. Envisaged applications include various forms of approximate reasoning. The starting point of this study is the extension of the classical concept of interpolation from the classical binary to the many-valued graded context. In any L-entailment system, given a commutative square of signature morphisms

and finite sets *E*<sup>1</sup> ⊆ *Sen*(Σ1) and *E*<sup>2</sup> ⊆ *Sen*(Σ2), we say that a finite set *E* ⊆ *Sen*(Σ) is a *Craig interpolant of E*<sup>1</sup> *and E*<sup>2</sup> when

$$(\theta\_1 E\_1 \vdash \theta\_2 E\_2 \; \; \le \; (E\_1 \vdash \varphi\_1 E) \; \* \; (\varphi\_2 E \vdash E\_2) . \tag{13}$$

When interpolants exist for all *E*1, *E*2, the respective commutative square of signature morphisms is called a *Craig interpolation square* (abbr. Ci square). When L is a residuated lattice, the concepts introduced in this definition extend also to L-institutions by considering the graded semantic entailment system.

In [109], there are some proper examples of the graded interpolation concept, proof theoretic as well as model theoretic. Some of the examples suggest that graded interpolation is much more subtle than the crips (binary truth) interpolation, as there are natural situations when crisp interpolation non-problems may be good graded interpolation problems.

Craig–Robinson interpolation [110] is an extended version of common (Craig) interpolation, this extension being especially relevant in computing science applications [44,65,70,83] but not only. In the binary case, under the presence of implication, the two versions of interpolation can be established as equivalent (an institution-independent proof can be found in [44]). In [109], this has been extended to graded interpolation under the assumption that L is a Heyting algebra and only for the graded semantic consequence relation in L-institutions.

Traditionally, model theoretic interpolation is causally related to Robinson consistency [2,111,112] and Beth definability [2,113]. These causalities have also been established in the abstract institution theoretic setting in [30,43,44]. Moreover, in [109], they have also been recovered at the many-valued truth level of L-institutions. However, that enterprise required a significant conceptual and mathematical effort that we will briefly and rather informally review in what follows.
