*2.3. Stratified Institutions*

Informally, the main idea behind the concept of stratified institution, as introduced in [5], is to enhance the concept of institution with "states" for the models. Thus, each model *M* comes equipped with a *set* [[*M*]] that has to satisfy some structural axioms. The following definition has been given in [6] and represents an important upgrade of the original definition from [5], the main reason being to make the definition of stratified institutions really usable for doing in-depth model theory. The latter has suffered another different upgrade in [7], which is, however, strongly convergent to the upgrade proposed in [6].

A *stratified institution* S is a tuple

$$(\operatorname{Sign}^{\mathcal{S}}, \operatorname{Sen}^{\mathcal{S}}, \operatorname{Mod}^{\mathcal{S}}, [\\_]^{\mathcal{S}}, | =^{\mathcal{S}})$$

consisting of:


$$\rho \operatorname{Mod}^{\mathcal{S}}(\varphi) \mathcal{M}' \ (\mathop{\vdash}^{\mathcal{S}})^{\parallel M' \parallel\_{\varphi} \mathcal{w}}\_{\Sigma} \text{ of and only if } \ M' \ (\mathop{\vdash}^{\mathcal{S}})^{\mathcal{w}}\_{\Sigma'} \operatorname{Sen}^{\mathcal{S}}(\varphi) \rho \tag{2}$$

holds for any signature morphism *ϕ*, *M* ∈ |*Mod*<sup>S</sup> (*ϕ*✷)|, *w* ∈ [[*M* ]]S *<sup>ϕ</sup>*✷, *ρ* ∈ *Sen*<sup>S</sup> (✷*ϕ*). Like for ordinary institutions, when appropriate, we also use simplified notations without superscripts or subscripts that are clear from the context.

The lax natural transformation property of [[\_]] is depicted in the diagram below

with the following compositionality property for each Σ -model *M* :

$$\|\|M^{\prime\prime}\|\_{\left(\varphi;\varphi^{\prime}\right)} = \|M^{\prime\prime}\|\_{\varphi^{\prime}}; \|\text{Mod}(\varphi^{\prime})M^{\prime\prime}\|\_{\varphi}.\tag{3}$$

Moreover, the natural transformation property of each [[\_]]*<sup>ϕ</sup>* is given by the commutativity of the following diagram:

*M h* [[*M* ]]Σ [[*M* ]]*ϕ* [[*h* ]]Σ [[*Mod*(*ϕ*)*M* ]]Σ [[*Mod*(*ϕ*)*h* ]]Σ *N* [[*N* ]]Σ [[*N* ]]*ϕ* [[*Mod*(*ϕ*)*<sup>N</sup>* ]]Σ (4)

The satisfaction relation can be presented as a natural transformation |= : *Sen* ⇒ [[*Mod*(\_) → **Set**]], where the functor [[*Mod*(\_) → **Set**]] : *Sign* → **Set** is defined by


*ϕ*

$$\mathbb{E}\left[\text{Mod}(\varphi)\to\mathbf{Set}\right](f)(M')=\left[\mathbb{M}'\right]\_{\mathfrak{g}}^{-1}(f(\text{Mod}(\mathfrak{g})(M')))\dots$$

A straightforward check reveals that the satisfaction condition (2) appears exactly as the naturality property of |=:

$$\begin{array}{l} \Sigma \\ \begin{array}{l} \\ \\ \Sigma' \end{array} \end{array} \xrightarrow{\operatorname{Sen}(\Sigma)} \begin{array}{l} \operatorname{Sen}(\Sigma) \xrightarrow{\left\| \begin{array}{l} \operatorname{Mod}(\Sigma) \to \ \mathbf{Set} \end{array} \right\|} \end{array} \begin{array}{l} \left\| \operatorname{Mod}(\Sigma) \to \ \mathbf{Set} \right\| \end{array}$$

$$\begin{array}{l} \Sigma' \\ \end{array} \begin{array}{l} \operatorname{Sen}(\Sigma') \xrightarrow{\left\| \begin{array}{l} \operatorname{Mod}(\varphi) \to \mathbf{Set} \end{array} \end{array} \begin{array}{l} \left\| \begin{array}{l} \operatorname{Mod}(\varphi) \to \mathbf{Set} \end{array} \right\| \end{array} \end{array} \begin{array}{l} \left\| \begin{array}{l} \operatorname{Mod}(\varphi) \to \mathbf{Set} \end{array} \right\| \end{array}$$

Ordinary institutions are the stratified institutions for which [[*M*]]<sup>Σ</sup> is always a singleton set. In the upgraded definition, we have removed the surjectivity condition on [[*M* ]]*ϕ* from the definition of the stratified institutions of [5] and rather make it explicit when necessary. This is motivated by the fact that most of the results developed do not depend upon this condition which, however, holds in all examples known by us. On the one hand, when modelling Kripke semantics abstractly, [[*M* ]]*<sup>ϕ</sup>* are even identities, which makes [[\_]] a strict rather than a lax natural transformation. However, on the other hand, there are interesting examples when the stratification is properly lax. One such example is provided by the representation result of this paper.

The following very expected property does not follow from the axioms of stratified institutions, hence we impose it explicitly.

**Assumption 1.** *In all considered stratified institutions, the satisfaction is preserved by model isomorphisms, i.e., for each* Σ*-model isomorphism h* : *M* → *N, each w* ∈ [[*M*]]Σ*, and each* Σ*-sentence ρ,*

$$\mathcal{M} \vdash^{\varpi} \rho \text{ if and only if } \mathcal{N} \vdash^{\|h\| \varpi} \rho.$$

The literature on stratified institutions shows many model theories that are captured as stratified institutions. Here, we recall some of them in a very succinct form; in a more detailed form, one may find them in [6,14].


7. Various kinds of automata theories can be presented as stratified institutions. For instance, the deterministic automata (for regular languages) have the set of the input symbols as signatures, the automata *A* are the models and the words are the sentences. Then, [[*A*]] is the set of the states of *<sup>A</sup>* and *<sup>A</sup>* <sup>|</sup>=*<sup>s</sup> <sup>α</sup>* if and only if *<sup>α</sup>* is recognized by *<sup>A</sup>* from the states *s*.
