Representing 3/2-Institutions as Stratified Institutions

Abstract

On the one hand, the extension of ordinary institution theory, known as the theory of stratified institutions, is a general axiomatic approach to model theories where the satisfaction is parameterized by states of the models. On the other hand, the theory of $3/2$-institutions is an extension of ordinary institution theory that accommodates the partiality of the signature morphisms and its syntactic and semantic effects. The latter extension is motivated by applications to conceptual blending and software evolution. In this paper, we develop a general representation theorem of $3/2$-institutions as stratified institutions. This enables a transfer of conceptual infrastructure from stratified to $3/2$-institutions. We provide some examples in this direction.

1. Introduction

1.1. Stratified Institutions

Institution theory is a general axiomatic approach to model theory that was originally introduced in computing science by Goguen and Burstall [1]. In institution theory, all three components of logical systems—namely the syntax, the semantics, and the satisfaction relation between them—are treated fully abstractly by relying heavily on category theory. This approach has impacted significantly both theoretical computing science [2] and model theory as such [3] (Both mentioned monographs rather reflect the stage of development of institution theory and its applications at the moment they were published or even before that. In the meantime, a lot of additional important developments have already taken place. At this moment, the literature on institution theory and that around it has been developed over the course of four decades or so and is rather vast.) In computing science, the concept of institution has emerged as the most fundamental mathematical structure of logic-based formal specifications, a great deal of theory being developed at the general level of abstract institutions. In model theory, the institution theoretic approach meant an axiomatic-driven redesign of core parts of model theory at a new level of generality—namely that of abstract institutions—independent of any concrete logical system. institution theoretic approach Moreover, there is a strong interdependency between the two lines of developments.

The institution theoretic approach to model theory has also been refined in order to address directly some important nonclassical model theoretic aspects. One such direction is motivated by models with states, appear in myriad forms in computing science and logic. A typical important class of examples is given by the Kripke semantics (of modal logics), which itself comes in a wide variety of forms. Moreover, the concept of model with states goes beyond Kripke semantics, at least in its conventional acceptations. For instance, various automata theories provide another important class of examples. The institution theory answer to “models with states” is given by the theory of stratified institutions introduced in [4,5] and further developed or invoked in works such as [6,7,8], etc.

1.2. 3/2-Institutions

Although in mainstream institution theory signature morphisms are considered in their full generality, they are always implicitly assumed to be total. However, there are few contexts that on the one hand require partial translations between signatures, and on the other hand require an institution theoretic treatment. Two such contexts are conceptual blending [9,10,11] and software evolution [12]. In [13], we have developed an extension of the ordinary concept of institution [1,3] that accommodates implicitly partiality of the signature morphisms in order to constitute foundations for the above-mentioned application domains. This new structure is called $3/2$-institution, and mathematically, is significantly more complex than ordinary institutions. One way to develop the theory of $3/2$-institutions is by representing them in another institution theory that enjoys a higher level of development, and through such a representation to import concepts and results from there. With this paper, we take a few steps in this direction.

The semantic effect of the (implicit) partiality of the signature morphisms in $3/2$-institutions is that the reduct of a model with respect to a given signature morphism is a set of models rather than a single model. This goes at the heart of our representation of $3/2$-institutions as stratified institutions: in the representation the states of a model consists of the set of its reducts (with respect to a given signature morphism).

1.3. Summary and Contributions

• In a preliminary section, we review (1) the category theory required by our work, (2) the concept of institution, (3) the concept of stratified institution, and (4) the concept of $3/2$-institution. Examples are also discussed briefly.
• The main section of the paper defines a representation of $3/2$-institutions as stratified institutions. We prove the correctness of this representation, i.e., that it satisfies the axioms of a stratified institution.
• One consequence is a further representation to ordinary institution theory via the adjunction from stratified institutions to ordinary institutions defined in [14] (formerly presented as a mere representation in [6]). Another consequence is the import of concepts of semantic connectives. The last consequence that we develop is about the relationship between model amalgamation properties in the representation and in the original $3/2$-institution.

2. Preliminaries

2.1. Categories

In general, we stick to the established category theoretic terminology and notations, such as in [15]. However, unlike there, we prefer to use the diagrammatic notation for compositions of arrows in categories, i.e., if $f:A\to B$ and $g:B\to C$ are arrows, then $f;g$ denotes their composition. The domain of an arrow/morphism f is denoted by $\square f$ while its codomain is denoted by $f\square$. $\mathbf{S}\mathbf{et}$ denotes the category of sets and functions and $\mathbf{C}\mathbf{A}\mathbf{T}$ the “quasi-category” of categories and functors (this means it is bigger than a category since the hom-sets are classes rather than sets). The class of objects of a category $\mathbf{C}$ is denoted by $\left|\mathbf{C}\right|$, and its class of arrows simply by $\mathbf{C}$ (so by $f\in \mathbf{C}$ we mean that f is an arrow in $\mathbf{C}$).

The dual of a category $\mathbf{C}$ (obtained by formally reversing its arrows) is denoted by ${\mathbf{C}}^{\ominus }$.

The following functor from [13] extends the well-known power-set construction from sets to categories. Given a category $\mathbf{C}$, the power-set category$\mathcal{P}\mathbf{C}$ is defined as follows:

• $\left|\mathcal{P}\mathbf{C}\right|=\{A\mid A\subseteq |C\left|\right\}$ and $\mathcal{P}\mathbf{C}(A,B)=\{H\subseteq \mathbf{C}\mid \square h\in A,h\square \in Bforeachh\in H\}$;
• Composition is defined by $H_1;H_2=\{h_1;h_2\mid h_1\in H_1,h_2\in H_2,h_1\square =\square h_2\}$; then $1_A=\{1_a\mid a\in A\}$ are the identities.

A partial function $f:A⇸B$ is a binary relation $f\subseteq A\times B$ such that $(a,b),(a,b^{\prime })\in f$ implies $b=b^{\prime }$. The definition domain of f, denoted $\mathrm{dom}\left(f\right)$ is the set $\{a\in A\mid \exists b(a,b)\in f\}$. A partial function $f:A⇸B$ is called total when $\mathrm{dom}\left(f\right)=A$. We denote by $f^0$ the restriction of f to $\mathrm{dom}\left(f\right)\times B$; this is a total function. Partial functions yield a subcategory of the category of binary relations, denoted $\mathbf{P}\mathbf{f}\mathbf{n}$. Note that $\mathrm{dom}(f;g)=\{a\in \mathrm{dom}\left(f\right)\mid f^0\left(a\right)\in \mathrm{dom}\left(g\right)\}$. If $A^{\prime }\subseteq A$ by $f\left(A^{\prime }\right)$, we denote the set $\{b\mid \exists a\in A^{\prime },(a,b)\in f\}$. Then, $f\left(A\right)$ is denoted by $\mathrm{Im}\left(f\right)$. It is easy to check the following (though not as immediate as in the case of the total functions): given partial functions $f:A⇸B$ and $g:B⇸C$ and $A^{\prime }\subseteq A$, we have that $(f;g)\left(A^{\prime }\right)=g\left(f\left(A^{\prime }\right)\right)$.

A $3/2$-category is just a category such that its hom-sets are partial orders, and the composition preserves these partial orders. In the literature, $3/2$-categories are also called ordered categories or locally ordered categories. In terms of enriched category theory [16], $3/2$-category are just categories enriched by the monoidal category of partially ordered sets.

Given a $3/2$-category $\mathbf{C}$ by ${\mathbf{C}}^{⦶}$, we denote its “vertical” dual which reverses the partial orders, and by ${\mathbf{C}}^{\oplus }$ its double dual ${\mathbf{C}}^{\ominus ⦶}$. Given $3/2$-categories $\mathbf{C}$ and ${\mathbf{C}}^{\prime }$, a strict $3/2$-functor $F:\mathbf{C}\to {\mathbf{C}}^{\prime }$ is a functor $\mathbf{C}\to {\mathbf{C}}^{\prime }$ that preserves the partial orders on the hom-sets. Lax functors relax the functoriality conditions $F\left(h\right);F\left(h^{\prime }\right)=F(h;h^{\prime })$ to $F\left(h\right);F\left(h^{\prime }\right)\le F(h;h^{\prime })$ (when $h\square =\square h^{\prime }$) and $F\left(1_A\right)=1_{F\left(A\right)}$ to $1_{F\left(A\right)}\le F\left(1_A\right)$. If these inequalities are reversed, then F is an oplax functor. This terminology complies to [17] and to more recent literature, but in earlier literature [18,19] this is reversed. Note that oplax + lax = strict. In what follows, whenever we say “$3/2$-functor” without the qualification “lax” or “oplax” we mean a functor which is either lax or oplax.

Lax functors can be composed like ordinary functors; we denote by $3/2\mathbf{C}\mathbf{A}\mathbf{T}$ the category of $3/2$-categories and lax functors.

Most typical examples of a $3/2$-category are $\mathbf{P}\mathbf{f}\mathbf{n}$—the category of partial functions in which the ordering between partial functions $A⇸B$ is given by the inclusion relation on the binary relations $A\to B$ and $\mathbf{P}\mathbf{o}\mathbf{S}\mathbf{E}\mathbf{T}$—the category of partially ordered sets (with monotonic mappings as arrows) with orderings between monotonic functions being defined point-wise ($f\le g$ if and only if $f\left(p\right)\le g\left(p\right)$ for all p).

The following $3/2$-category of [13] is instrumental for the concept of $3/2$-institution. The category $\mathbf{C}\mathbf{A}{\mathbf{T}}_{\mathcal{P}}$ has categories as objects and has arrows/morphisms $\mathbf{C}\to {\mathbf{C}}^{\prime }$ as mappings $\mathbf{C}\to \mathcal{P}{\mathbf{C}}^{\prime }$. The composition in $\mathbf{C}\mathbf{A}{\mathbf{T}}_{\mathcal{P}}$ is defined as follows: given $F:\mathbf{C}\to {\mathbf{C}}^{\prime }$ and $F^{\prime }:{\mathbf{C}}^{\prime }\to {\mathbf{C}}^{″}$ in $\mathbf{C}\mathbf{A}{\mathbf{T}}_{\mathcal{P}}$, then their composition is the mapping $\mathbf{C}\to \mathcal{P}{\mathbf{C}}^{″}$ that maps each arrow $f\in \mathbf{C}$ to the set ${\bigcup }_{f^{\prime }\in Ff}{F}^{\prime }{f}^{\prime }$.

By considering the point-wise partial order on the class of the mappings $\mathbf{C}\to \mathcal{P}{\mathbf{C}}^{\prime }$, we obtain a $3/2$-category denoted $3/2\left(\mathbf{C}\mathbf{A}{\mathbf{T}}_{\mathcal{P}}\right)$. Note that in the above definition, we do not require that the mappings $\mathbf{C}\to \mathcal{P}{\mathbf{C}}^{\prime }$ are functors of any kind, not even morphisms of graphs, they are just mappings between classes of arrows. In fact, the above composition in general does not preserve functoriality properties.

2.2. Institutions

The original standard reference for institution theory is [1]. An institution

$$\mathcal{I}=({\mathit{Sign}}^{\mathcal{I}},{\mathit{Sen}}^{\mathcal{I}},{\mathit{Mod}}^{\mathcal{I}},{\models }^{\mathcal{I}})$$

consists of:

• A category ${\mathit{Sign}}^{\mathcal{I}}$ whose objects are called signatures.
• A sentence functor ${\mathit{Sen}}^{\mathcal{I}}:{\mathit{Sign}}^{\mathcal{I}}\to \mathbf{S}\mathbf{et}$ defining for each signature a set whose elements are called sentences over that signature and defining for each signature morphism a sentence translation function.
• A model functor ${\mathit{Mod}}^{\mathcal{I}}:{\left({\mathit{Sign}}^{\mathcal{I}}\right)}^{\ominus }\to \mathbf{C}\mathbf{A}\mathbf{T}$ defining for each signature $\Sigma$ the category ${\mathit{Mod}}^{\mathcal{I}}(\Sigma )$ of Σ-models and $\Sigma$-model homomorphisms, and for each signature morphism $\phi$ the reduct functor ${\mathit{Mod}}^{\mathcal{I}}\left(\phi \right)$.
• For every signature $\Sigma$, a binary Σ-satisfaction relation ${\models }_{\Sigma }^{\mathcal{I}}\subseteq \left|{\mathit{Mod}}^{\mathcal{I}}(\Sigma )\right|\times {\mathit{Sen}}^{\mathcal{I}}(\Sigma )$.

Such that for each morphism $\phi$, the satisfaction condition

$$M^{\prime }{\models }_{{\Sigma }^{\prime }}^{\mathcal{I}}{\mathit{Sen}}^{\mathcal{I}}\left(\phi \right)\rho \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f} {\mathit{Mod}}^{\mathcal{I}}\left(\phi \right)M^{\prime }{\models }_{\Sigma }^{\mathcal{I}}\rho$$

(1)

holds for each $M^{\prime }\in \left|{\mathit{Mod}}^{\mathcal{I}}(\phi \square )\right|$ and $\rho \in {\mathit{Sen}}^{\mathcal{I}}(\square \phi )$. This can be expressed as the satisfaction relation ⊨ being a natural transformation:

($\left[\right|\mathit{Mod}(\Sigma )|\to 2]$ represents the “set” of the “subsets” of $\left|\mathit{Mod}\right(\Sigma \left)\right|$).

We may omit the superscripts or subscripts from the notations of the components of institutions when there is no risk of ambiguity. For example, if the considered institution and signature are clear, we may denote ${\models }_{\Sigma }^{\mathcal{I}}$ just by ⊨. For $M=\mathit{Mod}\left(\phi \right)M^{\prime }$, we say that M is the φ-reduct of $M^{\prime }$. The institution is called discrete when the model categories $\mathit{Mod}(\Sigma )$ are discrete (i.e., do not posses nonidentity arrows).

The literature (e.g., [2,3]) shows myriads of logical systems from computing or from mathematical logic captured as institutions. In fact, an informal thesis underlying institution theory is that any “logic” may be captured by the above definition. While this should be taken with a grain of salt, it certainly applies to any logical system based on satisfaction between sentences and models of any kind.

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$\left[\right[M\left]\right]$ 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 $\mathcal{S}$ is a tuple

$$({\mathit{Sign}}^{\mathcal{S}},{\mathit{Sen}}^{\mathcal{S}},{\mathit{Mod}}^{\mathcal{S}},{\left[[\_]\right]}^{\mathcal{S}},{\models }^{\mathcal{S}})$$

consisting of:

• Category ${\mathit{Sign}}^{\mathcal{S}}$ of signatures;
• A sentence functor ${\mathit{Sen}}^{\mathcal{S}}:{\mathit{Sign}}^{\mathcal{S}}\to \mathbf{S}\mathbf{et}$;
• A model functor ${\mathit{Mod}}^{\mathcal{S}}:{\left({\mathit{Sign}}^{\mathcal{S}}\right)}^{\ominus }\to \mathbf{C}\mathbf{A}\mathbf{T}$;
• A “stratification” lax natural transformation ${\left[[\_]\right]}^{\mathcal{S}}:{\mathit{Mod}}^{\mathcal{S}}\Rightarrow \mathit{SET}$, where $\mathit{SET}:{\mathit{Sign}}^{\mathcal{S}}\to \mathbf{C}\mathbf{A}\mathbf{T}$ is a functor mapping each signature to $\mathbf{S}\mathbf{et}$;
• A satisfaction relation between models and sentences which is parameterized by model states, $M{\left({\models }^{\mathcal{S}}\right)}_{\Sigma }^w\rho$, where $w\in {\left[\left[M\right]\right]}_{\Sigma }^{\mathcal{S}}$ such that the following satisfaction condition

  $${\mathit{Mod}}^{\mathcal{S}}\left(\phi \right)M^{\prime }{\left({\models }^{\mathcal{S}}\right)}_{\Sigma }^{{\displaystyle {\left[\left[M^{\prime }\right]\right]}_{\phi }}w}\rho \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f} M^{\prime }{\left({\models }^{\mathcal{S}}\right)}_{{\Sigma }^{\prime }}^w{\mathit{Sen}}^{\mathcal{S}}\left(\phi \right)\rho$$

  (2)

  holds for any signature morphism $\phi$, $M^{\prime }\in \left|{\mathit{Mod}}^{\mathcal{S}}(\phi \square )\right|$, $w\in {\left[\left[M^{\prime }\right]\right]}_{\phi \square }^{\mathcal{S}}$, $\rho \in {\mathit{Sen}}^{\mathcal{S}}(\square \phi )$.

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 $\left[\right[\_\left]\right]$ is depicted in the diagram below

with the following compositionality property for each ${\Sigma }^{″}$-model $M^{″}$:

$${\left[\left[M^{″}\right]\right]}_{(\phi ;{\phi }^{\prime })}={\left[\left[M^{″}\right]\right]}_{{\phi }^{\prime }};{\left[\left[\mathit{Mod}\left({\phi }^{\prime }\right)M^{″}\right]\right]}_{\phi }.$$

(3)

Moreover, the natural transformation property of each ${\left[[\_]\right]}_{\phi }$ is given by the commutativity of the following diagram:

(4)

The satisfaction relation can be presented as a natural transformation $\models :\mathit{Sen}\Rightarrow \left[\right[\mathit{Mod}(\_)\to \mathbf{S}\mathbf{et}\left]\right]$, where the functor $\left[\right[\mathit{Mod}(\_)\to \mathbf{S}\mathbf{et}\left]\right]:\mathit{Sign}\to \mathbf{S}\mathbf{et}$ is defined by

–

For each signature $\Sigma \in \left|\mathit{Sign}\right|$, $\left[\right[\mathit{Mod}(\Sigma )\to \mathbf{S}\mathbf{et}\left]\right]$ denotes the set of all the mappings $f:\left|\mathit{Mod}\right(\Sigma \left)\right|\to \mathbf{S}\mathbf{et}$ such that $f\left(M\right)\subseteq {\left[\left[M\right]\right]}_{\Sigma }$;

–

For each signature morphism $\phi :\Sigma \to {\Sigma }^{\prime }$

$$\left[[\mathit{Mod}\left(\phi \right)\to \mathbf{S}\mathbf{et}]\right]\left(f\right)\left(M^{\prime }\right)={\left[\left[M^{\prime }\right]\right]}_{\phi }^{-1}\left(f\left(\mathit{Mod}\left(\phi \right)\left(M^{\prime }\right)\right)\right).$$

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

Ordinary institutions are the stratified institutions for which ${\left[\left[M\right]\right]}_{\Sigma }$ is always a singleton set. In the upgraded definition, we have removed the surjectivity condition on ${\left[\left[M^{\prime }\right]\right]}_{\phi }$ 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, ${\left[\left[M^{\prime }\right]\right]}_{\phi }$ are even identities, which makes $\left[\right[\_\left]\right]$ 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 $h:M\to N$, each $w\in {\left[\left[M\right]\right]}_{\Sigma }$, and each Σ-sentence ρ,

$$M{\models }^w\rho if and only ifN{\models }^{\left[\right[h\left]\right]w}\rho .$$

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].

• In modal propositional logic ($\mathcal{M}\mathcal{P}\mathcal{L}$), the category of the signatures is $\mathbf{S}\mathbf{et}$; $\mathit{Sen}\left(P\right)$ is the set of the usual modal sentences formed with the atomic propositions from P, and the P-models are the Kripke structures $(W,M)$, where $W=\left(\right|W|,W_{\lambda })$ consists of a set of “possible worlds” $\left|W\right|$ and an accessibility relation $W_{\lambda }\subseteq \left|W\right|\times \left|W\right|$, and $M:\left|W\right|\to 2^P$. The stratification is given by $\left[\right[(W,M)\left]\right]=\left|W\right|$.
• In first-order modal logic ($\mathcal{M}\mathcal{F}\mathcal{O}\mathcal{L}$), the signatures are first-order logic ($\mathcal{F}\mathcal{O}\mathcal{L}$) signatures consisting of sets of operation and relation symbols structured by their arities. The sentences extend the usual construction of $\mathcal{F}\mathcal{O}\mathcal{L}$ sentences with the modal connectives □ and ⋄. The models for a signature $\Sigma$ are Kripke structures $(W,M)$, where W is like in $\mathcal{M}\mathcal{P}\mathcal{L}$ but $M:\left|W\right|\to |{\mathit{Mod}}^{\mathcal{F}\mathcal{O}\mathcal{L}}(\Sigma )|$ subject to the constraint that the carrier sets and the interpretations of the constants are shared across the possible worlds. The stratification is like in $\mathcal{M}\mathcal{P}\mathcal{L}$.
• Hybrid logics refine modal logics by adding explicit syntax for the possible worlds such as nominals and @. Institutions of hybrid logics upgrade the syntactic and the semantic components of the institutions of modal logics accordingly.
• Multimodal logics exhibit several modalities instead of only the traditional ⋄ and □ and, moreover, these may have various arities. If one considers the sets of modalities to be variable, then they have to be considered as part of the signatures. Each of the stratified institutions discussed in the previous examples admit an upgrade to the multimodal case.
• In a series of works on modalization of institutions, Refs. [20,21,22] modal logic and Kripke semantics are developed by abstracting away details that do not belong to modality, such as sorts, functions, predicates, etc. This is achieved by extensions of abstract institutions (in the standard situations meant in principle to encapsulate the atomic part of the logics) with the essential ingredients of modal logic and Kripke semantics. The result of this process, when instantiated to various concrete logics (or to their atomic parts only), generates uniformly a wide range of hierarchical combinations between various flavors of modal logic and various other logics. Concrete examples discussed in [20,21,22] include various modal logics over nonconventional structures of relevance in computing science, such as partial algebra, preordered algebra, etc. Various constraints on the respective Kripke models, many of them having to do with the underlying nonmodal structures, have also been considered. All these arise as examples of stratified institutions, such as the examples presented above in the paper. An interesting class of examples that has emerged quite smoothly out of the general works on hybridization (i.e., modalization including also hybrid logic features) of institutions is that of multilayered hybrid logics that provide a logical base for specifying hierarchical transition systems (see [23]).
• Open first order logic ($\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}$). This is a $\mathcal{F}\mathcal{O}\mathcal{L}$ instance of $St\left(\mathcal{I}\right)$, the “internal stratification” abstract example developed in [5]. An $\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}$ signature is a pair $(\Sigma ,X)$ consisting of $\mathcal{F}\mathcal{O}\mathcal{L}$ signature $\Sigma$ and a finite block of variables. To any $\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}$ signature $(\Sigma ,X)$, there corresponds a $\mathcal{F}\mathcal{O}\mathcal{L}$ signature $\Sigma +X$ that adjoins X to $\Sigma$ as new constants. Then, ${\mathit{Sen}}^{\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}}(\Sigma ,X)={\mathit{Sen}}^{\mathcal{F}\mathcal{O}\mathcal{L}}(\Sigma +X)$, ${\mathit{Mod}}^{\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}}(\Sigma ,X)={\mathit{Mod}}^{\mathcal{F}\mathcal{O}\mathcal{L}}(\Sigma )$, ${\left[\left[M\right]\right]}_{\Sigma ,X}=M^X$, i.e., the set of the “valuations” of X to M and for each $(\Sigma ,X)$-model M, each $w\in M^X$, and each $(\Sigma ,X)$-sentence $\rho$, we define $\left(M{\left({\models }_{\Sigma ,X}^{\mathcal{O}\mathcal{F}\mathcal{O}\mathcal{L}}\right)}^w\rho \right)=\left(M^w{\models }_{\Sigma +X}^{\mathcal{F}\mathcal{O}\mathcal{L}}\rho \right)$, where $M^w$ is the expansion of M to $\Sigma +X$ such that $M_X^w=w$ (i.e., the new constants of X are interpreted in $M^w$ according to the “valuation” w).
• 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, $\left[\right[A\left]\right]$ is the set of the states of A and $A{\models }^s\alpha$ if and only if $\alpha$ is recognized by A from the states s.

2.4. $3/2$-Institutions

The concept of $3/2$-institution has been introduced in [13]. Our presentation of $3/2$-institutions follows that paper. A $3/2$-institution $\mathcal{I}=({\mathit{Sign}}^{\mathcal{I}},{\mathit{Sen}}^{\mathcal{I}},{\mathit{Mod}}^{\mathcal{I}},{\models }^{\mathcal{I}})$ consists of

• A $3/2$-category ${\mathit{Sign}}^{\mathcal{I}}$—called the category of the signatures;
• A $3/2$-functor ${\mathit{Sen}}^{\mathcal{I}}:{\mathit{Sign}}^{\mathcal{I}}\to \mathbf{P}\mathbf{f}\mathbf{n}$—called the sentence functor;
• A lax $3/2$-functor ${\mathit{Mod}}^{\mathcal{I}}:{\left({\mathit{Sign}}^{\mathcal{I}}\right)}^{\oplus }\to 3/2\left(\mathbf{C}\mathbf{A}{\mathbf{T}}_{\mathcal{P}}\right)$ — called the model functor, such that $\mathit{Mod}\left(\phi \right)$ is a lax functor for each signature morphism $\phi$;
• For each signature $\Sigma \in |{\mathit{Sign}}^{\mathcal{I}}|$, a satisfaction relation ${\models }_{\Sigma }^{\mathcal{I}}\subseteq \left|{\mathit{Mod}}^{\mathcal{I}}(\Sigma )\right|\times {\mathit{Sen}}^{\mathcal{I}}(\Sigma )$.

Such that for each morphism $\phi \in {\mathit{Sign}}^{\mathcal{I}}$, the satisfaction condition

$$M^{\prime }{\models }_{\phi \square }^{\mathcal{I}}{\mathit{Sen}}^{\mathcal{I}}\left(\phi \right)\rho \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f}M{\models }_{\square \phi }^{\mathcal{I}}\rho$$

(5)

holds for each $M^{\prime }\in \left|{\mathit{Mod}}^{\mathcal{I}}(\phi \square )\right|$, each $M\in |{\mathit{Mod}}^{\mathcal{I}}\left(\phi \right)M^{\prime }|$, and each $\rho \in \mathrm{dom}\left({\mathit{Sen}}^{\mathcal{I}}\left(\phi \right)\right)$.

The difference between $3/2$-institutions and ordinary institutions, from now on called 1-institutions, is determined by the $3/2$-categorical structure of the signature morphisms which propagates to the sentence and to the model functors. Consequently, the satisfaction condition (5) takes an appropriate format. Thus, for each signature morphism $\phi$, its corresponding sentence translation $\mathit{Sen}\left(\phi \right)$ is a partial function $\mathit{Sen}(\square \phi )\to \mathit{Sen}(\phi \square )$ and, moreover, whenever $\phi \le \theta$, we have that $\mathit{Sen}\left(\phi \right)\subseteq \mathit{Sen}\left(\theta \right)$. The sentence functor $\mathit{Sen}$ can be either lax or oplax, and depending on how this is, we may call the respective $3/2$-institution a lax or oplax $3/2$-institution. In many concrete situations, it happens that $\mathit{Sen}$ is strict, while some general results require it to be either lax, oplax or strict.

The model reduct $\mathit{Mod}\left(\phi \right)$ is a lax functor $\mathit{Mod}(\phi \square )\to \mathcal{P}\mathit{Mod}(\square \phi )$, implying that for each ${\Sigma }^{\prime }$-model $M^{\prime }$ we have a class of reductsM rather than a single reduct. In concrete examples, this is a direct consequence of the partiality of $\phi$: in the reducts, the interpretation of the symbols on which $\phi$ is not defined is unconstrained, therefore there may be many possibilities for their interpretations. “Many” here includes also the case when there is no interpretation.

–

The fact that $\mathit{Mod}$ is a $3/2$-functor implies also that whenever $\phi \le \theta$ we have $\mathit{Mod}\left(\theta \right)\le \mathit{Mod}\left(\phi \right)$, i.e., $\mathit{Mod}\left(\theta \right)M^{\prime }\subseteq \mathit{Mod}\left(\phi \right)M^{\prime }$, etc.

–

The lax aspect of $\mathit{Mod}$ means that for signature morphisms $\phi$ and ${\phi }^{\prime }$, such that $\phi \square =\square {\phi }^{\prime }$, and for any ${\phi }^{\prime }\square$-model $M^{″}$, we have that

$$\mathit{Mod}\left(\phi \right)\left(\mathit{Mod}\left({\phi }^{\prime }\right)M^{″}\right)\subseteq \mathit{Mod}(\phi ;{\phi }^{\prime })M^{″}$$

and for each signature $\Sigma$ and for each $\Sigma$-model M that

$$M\in \mathit{Mod}\left(1_{\Sigma }\right)M.$$

–

The lax aspect of the reduct functors $\mathit{Mod}\left(\phi \right)$ means that for model homomorphisms $h_1,h_2$, such that $h_1\square =\square h_2$, we have that

$$\mathit{Mod}\left(\phi \right)\left(h_1\right);\mathit{Mod}\left(\phi \right)\left(h_2\right)\subseteq \mathit{Mod}\left(\phi \right)(h_1;h_2)$$

and for each $M^{\prime }\in \mathit{Mod}(\phi \square )$ and each $M\in \mathit{Mod}\left(\phi \right)M^{\prime }$ that

$$1_M\in \mathit{Mod}\left(\phi \right)1_{M^{\prime }}.$$

The model homomorphisms do not yet play any role in conceptual blending or in other envisaged applications of $3/2$-institutions. Hence, the lax aspect of model functors is for the moment a purely theoretical feature which is, however, supported naturally by all examples. Another technical note: according to the definition of $3/2$-institutions. At the abstract level, there are several implicit ways to consider the “totality” of signature morphisms in terms of their syntactic and semantic effects. The following concepts have been introduced in [13]. A signature morphism $\phi$ in a $3/2$-institution

• Is $\mathit{Sen}$-maximal when $\mathit{Sen}\left(\phi \right)$ is total;
• Is $\mathit{Mod}$-maximal when for each $\phi \square$-model $M^{\prime }$, $\mathit{Mod}\left(\phi \right)M^{\prime }$ is a singleton;
• Is total when it is both $\mathit{Sen}$-maximal and $\mathit{Mod}$-maximal;
• Is $\mathit{Mod}$-strict when for each signature morphism $\theta$, such that $\theta \square =\square \phi$, we have that

  $$\mathit{Mod}\left(\phi \right);\mathit{Mod}\left(\theta \right)=\mathit{Mod}(\theta ;\phi ).$$

In general, in many concrete situations of interest, a signature morphism is $\mathit{Mod}$-strict whenever it is total. In [13], there is even a result of a general nature that supports this fact.

The seminal paper [13] presents in detail a series of examples of $3/2$-institutions. Here, we recall from there three classes of examples in a very succinct form.

• Common examples of institutions can be turned into $3/2$-institutions by introducing explicit partiality at the level of the signature morphisms. This means that certain sort/operation/relation symbols are skipped by the respective (partial) signature morphism. This induces a further partiality on the sentence translations as only the sentences that does not contain “skipped” symbols can be translated. The model reducts may interpret freely the “skipped” symbols, hence in principle one model may have several reducts along the same signature morphisms. In [13], this procedure was illustrated on the institutions of propositional logic (hence $3/2\mathcal{P}\mathcal{L}$) and of many-sorted algebra (hence $3/2\mathcal{M}\mathcal{S}\mathcal{A}$). In the latter case, several degrees of partiality can be introduced.
• The $3/2$-institutional seeds of [13] provide a generic way to define $3/2$-institutions. Some of the $3/2$-institution that are based on some form of explicit partiality can also be presented in this way.
• We may turn any abstract institution into a proper $3/2$-institution by adding weights to the signature morphisms, which means that the signature morphisms come as pairs $(\phi ,k)$ (in [13] denoted $\phi •k$), where $\phi$ is a signature morphism of the ordinary institution and $k\in \{0,1\}$. The signatures stay the same. This construction is extended to sentences and models in a way that yields the sentence and the model functors proper lax functors. Although this is a mere technical construction without any known applications, it has an important theoretical significance because it provides a class of examples where the $3/2$-categorical structures involved have nothing to do with any form of partiality.

3. The Canonical Stratified Institution Associated to a $3/2$-Institution

The representation of $3/2$-institutions as stratified institutions is in general partial in the sense that the signature morphisms that are subject to the representation have to satisfy certain technical conditions. Two of these are defined below. The second one appears as a $3/2$-institution theoretic replica of a property from ordinary institution theory [3] with the same name but in a somewhat reverse form. While the former concept is a lifting concept, the $3/2$-institution theoretic one may have an opposite appearance because it goes along the direction of the model reduct. However, this is misleading because in $3/2$-institutions, due to the implicit partiality of the signature morphisms, reducts also have a nature of expansion. Towards the end of this section, we discuss what these two properties mean in concrete situations. The constructions and the results in this section are developed at the abstract level. It would be helpful if the reader would interpret them in the context of the examples of $3/2$-institutions listed above. This should be a rather straightforward exercise, especially if one considers the discussion on the technical conditions at the end of this section.

Definition 1.

In any $3/2$-institution, a signature morphism χ

• Is fiber-small when for each $\chi \square$-model M we have that $\mathit{Mod}\left(\chi \right)M$ is a set;
• Is quasi-representable when for each $\chi \square$-model homomorphism $h:M\to M_0$, and for each $N\in \mathit{Mod}\left(\chi \right)M$ there exists and unique model homomorphism $h^N\in \mathit{Mod}\left(\chi \right)h$, such that $\square h^N=N$.

We now fix a $3/2$-institution $\mathcal{I}=(\mathit{Sign},\mathit{Sen},\mathit{Mod},\models )$ and gradually build the entities that define its associated stratified institution ${\mathcal{I}}^{\mathrm{s}}=({\mathit{Sign}}^{\mathrm{s}},{\mathit{Sen}}^{\mathrm{s}},{\mathit{Mod}}^{\mathrm{s}},\left[[\_]\right],{\models }^{\mathrm{s}})$. The main idea of this representation is that the reducts of a model M are considered to be its states. In order to make precise sense of this idea, we have to change the concept of signature: in the stratified institution, a signature is a certain signature morphism $\chi$ in $\mathcal{I}$, such that M is a $\chi \square$-model. It is the abstract nature of the concept of institution that allows for such a conceptual twist.

Definition 2

(The category of the signatures). The category ${\mathit{Sign}}^{\mathrm{s}}$ has the objects the fiber-small quasi-representable signature morphisms χ of $\mathit{Sign}$. The arrows $\chi \to {\chi }^{\prime }$ in ${\mathit{Sign}}^{\mathrm{s}}$ are pairs of signature morphisms $(\phi ,\theta )$, such that

• Both φ and θ are total and $\mathit{Mod}$-strict;
• $\chi ;\theta \le \phi ;{\chi }^{\prime }$.

  (6)

The composition in ${\mathit{Sign}}^{\mathrm{s}}$ is defined as pairwise composition in $\mathit{Sign}$, i.e., $(\phi ,\theta );({\phi }^{\prime },{\theta }^{\prime })=(\phi ;{\phi }^{\prime },\theta ;{\theta }^{\prime })$, as shown in the following diagram:

(7)

An arrow $(\phi ,\theta ):\chi \to {\chi }^{\prime }$ is strict when $\chi ;\theta =\phi ;{\chi }^{\prime }$.

We have the correctness of definition 1:

Proposition 1.

${\mathit{Sign}}^{\mathrm{s}}$ is a category.

Proof. 

We have to prove that the composition preserves the preorder property. This follows from the monotonicity of the composition in $\mathit{Sign}$ (we use the notations from (7)):

$$\chi ;\theta ;{\theta }^{\prime }\le \phi ;{\chi }^{\prime };{\theta }^{\prime }\le \phi ;{\phi }^{\prime };{\chi }^{\prime }.$$

It remains to note that totality and $\mathit{Mod}$-strictness when considered together are preserved by the composition of the signature morphisms. ($\mathit{Mod}$-strictness supports the preservation of $\mathit{Mod}$-maximality by the composition of signature morphisms.) □

Definition 3

(The sentence translation functor). For any ${\mathit{Sign}}^{\mathrm{s}}$ signature χ, we define ${\mathit{Sen}}^{\mathrm{s}}\left(\chi \right)=\mathit{Sen}(\square \chi )$ and for any ${\mathit{Sign}}^{\mathrm{s}}$-morphism $(\phi ,\theta )$, we define ${\mathit{Sen}}^{\mathrm{s}}(\phi ,\theta )=\mathit{Sen}\left(\phi \right)$.

Proposition 2.

${\mathit{Sen}}^{\mathrm{s}}$ is a functor ${\mathit{Sign}}^{\mathrm{s}}\to \mathbf{S}\mathbf{et}$.

Proof. 

This is an immediate consequence of the functoriality of $\mathit{Sen}$ and of the totality hypothesis, which guarantees that $\mathit{Sen}\left(\phi \right)$ is indeed a total function. □

Definition 4

(The model reduct functor). For any ${\mathit{Sign}}^{\mathrm{s}}$ signature χ, we define ${\mathit{Mod}}^{\mathrm{s}}\left(\chi \right)=\mathit{Mod}(\chi \square )$ and for any ${\mathit{Sign}}^{\mathrm{s}}$-morphism $(\phi ,\theta )$ we define ${\mathit{Mod}}^{\mathrm{s}}(\phi ,\theta )=\mathit{Mod}\left(\theta \right)$.

Proposition 3.

${\mathit{Mod}}^{\mathrm{s}}$ is a functor ${\left({\mathit{Sign}}^{\mathrm{s}}\right)}^{\ominus }\to \mathbf{C}\mathbf{A}\mathbf{T}$.

Proof. 

This is an immediate consequence of the functoriality of $\mathit{Mod}$ and of the $\mathit{Mod}$-maximality and $\mathit{Mod}$-strictness hypothesis (on $\theta$). □

Definition 5

(The stratification). For any ${\mathit{Sign}}^{\mathrm{s}}$ signature χ, we define

• ${\left[\left[M\right]\right]}_{\chi }=\mathit{Mod}\left(\chi \right)M$ for any $M\in |{\mathit{Mod}}^{\mathrm{s}}\left(\chi \right)\left|\right(=\left|\mathit{Mod}(\chi \square )\right|)$;
• ${\left[\left[h\right]\right]}_{\chi }N=h^N\square$ for any $\chi \square$-model homomorphism $h\in {\mathit{Mod}}^{\mathrm{s}}\left(\chi \right)(=|\mathit{Mod}(\chi \square )\left|\right)$ and $\square \chi$-model $N\in \mathit{Mod}\left(\chi \right)(\square h)$.

For each signature morphism $(\phi ,\theta ):\chi \to {\chi }^{\prime }$ in ${\mathit{Sign}}^{\mathrm{s}}$, we define:

• ${\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}{N}^{\prime }=\mathit{Mod}\left(\phi \right)N^{\prime }$ for any $M^{\prime }\in \left|{\mathit{Mod}}^{\mathrm{s}}\left({\chi }^{\prime }\right)\right|$ and any $N^{\prime }\in {\left[\left[M^{\prime }\right]\right]}_{{\chi }^{\prime }}$.

Proposition 4.

$\left[\right[\_\left]\right]$ is a lax natural transformation ${\mathit{Mod}}^{\mathrm{s}}\Rightarrow \mathit{SET}$.

Proof. 

The correctness of definition 5 is justified as follows:

• ${\left[\left[M\right]\right]}_{\chi }$ is a set by the fiber-small assumption on $\chi$.
• The definition of ${\left[\left[h\right]\right]}_{\chi }N$ relies on the quasi-representability assumption on $\chi$.
• ${\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}{N}^{\prime }$ represents a single model because $\phi$ is $\mathit{Mod}$-maximal.
• That ${\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}\left({\left[\left[M^{\prime }\right]\right]}_{{\chi }^{\prime }}\right)\subseteq {\left[\left[{\mathit{Mod}}^{\mathrm{s}}(\phi ,\theta )M^{\prime }\right]\right]}_{\chi }$ is shown as follows:

  $$\begin{array}{ccc}\hfill {\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}\left({\left[\left[M^{\prime }\right]\right]}_{{\chi }^{\prime }}\right)=& \mathit{Mod}\left(\phi \right)\left(\mathit{Mod}\left({\chi }^{\prime }\right)M^{\prime }\right)\hfill & \hfill \mathrm{definition}\mathrm{of}\left[\right[\_\left]\right]\\ \hfill \subseteq & \mathit{Mod}(\phi ;{\chi }^{\prime })M^{\prime }\hfill & \hfill \mathit{Mod}\mathrm{lax}\\ \hfill \subseteq & \mathit{Mod}(\chi ;\theta )M^{\prime }\hfill & \hfill \chi ;\theta \le \phi ;{\chi }^{\prime },\mathit{Mod}\mathrm{monotone}\\ \hfill =& \mathit{Mod}\left(\chi \right)\left(\mathit{Mod}\left(\theta \right)M^{\prime }\right)\hfill & \hfill \theta \mathit{Mod}-\mathrm{strict}\\ \hfill =& {\left[\left[\mathit{Mod}\left(\theta \right)M^{\prime }\right]\right]}_{\chi }\hfill & \hfill \mathrm{definition}\mathrm{of}{\left[[\_]\right]}_{\chi },\theta \mathit{Mod}-\mathrm{maximal}\\ \hfill =& {\left[\left[{\mathit{Mod}}^{\mathrm{s}}(\phi ,\theta )M^{\prime }\right]\right]}_{\chi }\hfill & \hfill \mathrm{definition}\mathrm{of}{\mathit{Mod}}^{\mathrm{s}}.\end{array}$$

The functoriality of ${\left[[\_]\right]}_{\chi }:{\mathit{Mod}}^{\mathrm{s}}\left(\chi \right)\to \mathbf{S}\mathbf{et}$ means two things:

• That ${\left[\left[1_M\right]\right]}_{\chi }N=N$ for each $\chi \square$-model M and each $N\in \mathit{Mod}\left(\chi \right)M$. This is shown by the following argument. Since $\mathit{Mod}\left(\chi \right)$ is lax, it follows that $1_N\in \mathit{Mod}\left(\chi \right)1_M$, which by the uniqueness aspect of the quasi-representability property implies that $1_N={\left(1_M\right)}^N$. Since $1_N\square =N$, the conclusion follows.
• That ${\left[[h;h_0]\right]}_{\chi }={\left[\left[h\right]\right]}_{\chi };{\left[\left[h_0\right]\right]}_{\chi }$ for any $\chi \square$-model homomorphisms $h:M\to M_0$ and $h_0:M_0\to M_1$. In order to show that, we consider any model $N\in \mathit{Mod}\left(\chi \right)M$ and denote by $N_0=h^N\square$. We have that:

  $$\begin{array}{ccc}\hfill h^N;h_0^{N_0}\in & \mathit{Mod}\left(\chi \right)h;\mathit{Mod}\left(\chi \right)h_0\hfill & \hfill \mathrm{definitions}\mathrm{of}{h}_0^N,h_0^{N_0}\\ \hfill \subseteq & \mathit{Mod}\left(\chi \right)(h;h_0)\hfill & \hfill \mathit{Mod}\left(\chi \right)\mathrm{lax}.\end{array}$$
  From the uniqueness aspect of the quasi-representability property of $\chi$, it follows that

  $$h^N;h_0^{N_0}={(h;h_0)}^N.$$

  (8)
  Hence,

  $$\begin{array}{ccc}\hfill {\left[[h;h_0]\right]}_{\chi }N=& {(h;h_0)}^N\square \hfill & \hfill \mathrm{definition}\mathrm{of}{\left[[h;h_0]\right]}_{\chi }N\\ \hfill =& (h^N;h_0^{N_0})\square \hfill & \hfill \left(8\right)\\ \hfill =& h_0^{N_0}\square \hfill & \\ \hfill =& {\left[\left[h_0\right]\right]}_{\chi }\left({\left[\left[h\right]\right]}_{\chi }N\right)\hfill & \hfill \mathrm{definitions}\mathrm{of}{N}_0,{\left[\left[h_0\right]\right]}_{\chi }{N}_0,{\left[\left[h\right]\right]}_{\chi }N.\end{array}$$

For proving the lax natural transformation property of $\left[\right[\_\left]\right]$ (relation ((3))), we consider a composition of signature morphisms in ${\mathit{Sign}}^{\mathrm{s}}$ such as in diagrams (7), an ${\Omega }^{″}$-model $M^{″}$, and $N^{″}\in {\left[\left[M^{″}\right]\right]}_{{\chi }^{″}}=\mathit{Mod}\left({\chi }^{″}\right)M^{″}$. Note that since $N^{″}\in {\left[\left[M^{″}\right]\right]}_{{\chi }^{″}}$, we have that

$$\mathit{Mod}\left({\phi }^{\prime }\right)N^{″}={\left[\left[M^{″}\right]\right]}_{({\phi }^{\prime },{\theta }^{\prime })}{N}^{″}\in {\left[\left[{\mathit{Mod}}^{\mathrm{s}}({\phi }^{\prime },{\theta }^{\prime })M^{″}\right]\right]}_{{\chi }^{\prime }}={\left[\left[\mathit{Mod}\left({\theta }^{\prime }\right)M^{″}\right]\right]}_{{\chi }^{\prime }}$$

(9)

Then, we have:

${\left[\left[M^{″}\right]\right]}_{(\phi ,\theta );({\phi }^{\prime },{\theta }^{\prime })}{N}^{″}=$

$$\begin{array}{ccc}\hfill =& {\left[\left[M^{″}\right]\right]}_{(\phi ;{\phi }^{\prime },\theta ;{\theta }^{\prime })}{N}^{″}\hfill & \hfill \mathrm{definition}\mathrm{of}{\mathit{Sign}}^{\mathrm{s}}\\ \hfill =& \mathit{Mod}(\phi ;{\phi }^{\prime })N^{″}\hfill & \hfill \mathrm{definition}\mathrm{of}\left[\right[\_\left]\right]\\ \hfill =& \mathit{Mod}\left(\phi \right)\left(\mathit{Mod}\left({\phi }^{\prime }\right)N^{″}\right)\hfill & \hfill \mathit{Mod}\mathrm{functor},\phi \mathit{Mod}-\mathrm{strict}\\ \hfill =& {\left[\left[\mathit{Mod}\left({\theta }^{\prime }\right)M^{″}\right]\right]}_{(\phi ,\theta )}\left(\mathit{Mod}\left({\phi }^{\prime }\right)N^{″}\right)\hfill & \hfill \left(9\right),\mathrm{definition}\mathrm{of}\left[\right[\_\left]\right]\\ \hfill =& {\left[\left[{\mathit{Mod}}^{\mathrm{s}}({\phi }^{\prime },{\theta }^{\prime })M^{″}\right]\right]}_{(\phi ,\theta )}\left(\mathit{Mod}\left({\phi }^{\prime }\right)N^{″}\right)\hfill & \hfill \mathrm{definition}\mathrm{of}{\mathit{Mod}}^{\mathrm{s}}\\ \hfill =& {\left[\left[{\mathit{Mod}}^{\mathrm{s}}({\phi }^{\prime },{\theta }^{\prime })M^{″}\right]\right]}_{(\phi ,\theta )}\left({\left[\left[M^{″}\right]\right]}_{({\phi }^{\prime },{\theta }^{\prime })}{N}^{″}\right)\hfill & \hfill \mathrm{definition}\mathrm{of}\left[\right[\_\left]\right].\end{array}$$

For proving the natural transformation property of ${\left[[\_]\right]}_{(\phi ,\theta )}$, under the notations from diagrams (7) by considering an ${\Omega }^{\prime }$-model homomorphism $h^{\prime }:M^{\prime }\to M_0^{\prime }$, we have to show that the diagram below commutes:

(10)

Let $N^{\prime }\in \mathit{Mod}\left({\chi }^{\prime }\right)M^{\prime }$, and let us denote $N=\mathit{Mod}\left(\phi \right)N^{\prime }$ and $h=\mathit{Mod}\left(\theta \right)h^{\prime }$. Let us first prove that

$$\mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}=h^N.$$

(11)

On the one hand, we have:

$$\begin{array}{ccc}\hfill \mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}\in & \mathit{Mod}\left(\phi \right)\left(\mathit{Mod}\left({\chi }^{\prime }\right)h^{\prime }\right)\hfill & \hfill \mathrm{definition}\mathrm{of}{h^{\prime }}^{N^{\prime }}\\ \hfill \subseteq & \mathit{Mod}(\phi ;{\chi }^{\prime })h^{\prime }\hfill & \hfill \mathit{Mod}\mathrm{lax}\\ \hfill \subseteq & \mathit{Mod}(\chi ;\theta )h^{\prime }\hfill & \hfill \mathit{Mod}\mathrm{monotone},\chi ;\theta \le \phi ;{\chi }^{\prime }\\ \hfill =& \mathit{Mod}\left(\chi \right)\left(\mathit{Mod}\left(\theta \right)h^{\prime }\right)\hfill & \hfill \theta \mathit{Mod}-\mathrm{strict}\\ \hfill =& \mathit{Mod}\left(\chi \right)h.\hfill \end{array}$$

On the other hand, we have:

$$\square \mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}=\mathit{Mod}\left(\phi \right)(\square {h^{\prime }}^{N^{\prime }})=\mathit{Mod}\left(\phi \right)N^{\prime }=N.$$

Then, (11) follows from $\mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}\in \mathit{Mod}\left(\chi \right)h$ and $\square \mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}=N$ and from the uniqueness aspect of the quasi-representability property of $\chi$. Now, the following argument completes the proof of the natural transformation property of ${\left[[\_]\right]}_{(\phi ,\theta )}$:

$$\begin{array}{ccc}\hfill {\left[\left[\mathit{Mod}\left(\theta \right)h^{\prime }\right]\right]}_{\chi }\left({\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}{N}^{\prime }\right)=& {\left[\left[\mathit{Mod}\left(\theta \right)h^{\prime }\right]\right]}_{\chi }\left(\mathit{Mod}\left(\phi \right)N^{\prime }\right)\hfill & \hfill \mathrm{definition}\mathrm{of}{\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}\\ \hfill =& h^N\square \hfill & \hfill \mathrm{definition}\mathrm{of}{\left[\left[\mathit{Mod}\left(\theta \right)h^{\prime }\right]\right]}_{\chi }={\left[\left[h\right]\right]}_{\chi }\\ \hfill =& \left(\mathit{Mod}\left(\phi \right){h^{\prime }}^{N^{\prime }}\right)\square \hfill & \hfill \left(11\right)\\ \hfill =& \mathit{Mod}\left(\phi \right)({h^{\prime }}^{N^{\prime }}\square )\hfill & \hfill \mathit{Mod}\left(\phi \right)\mathrm{functor}\\ \hfill =& {\left[\left[M_0^{\prime }\right]\right]}_{(\phi ,\theta )}({h^{\prime }}^{N^{\prime }}\square )\hfill & \hfill \mathrm{definition}\mathrm{of}{\left[\left[M_0^{\prime }\right]\right]}_{(\phi ,\theta )}\\ \hfill =& {\left[\left[M_0^{\prime }\right]\right]}_{(\phi ,\theta )}\left({\left[\left[h^{\prime }\right]\right]}_{{\chi }^{\prime }}{N}^{\prime }\right)\hfill & \hfill \mathrm{definition}\mathrm{of}{\left[\left[h^{\prime }\right]\right]}_{{\chi }^{\prime }}.\end{array}$$

□

Definition 6

(The satisfaction relation). For each signature χ in ${\mathit{Sign}}^{\mathrm{s}}$, each $\chi \square$-model M, each $\square \chi$-model $N\in {\left[\left[M\right]\right]}_{\chi }$, and each $\square \chi$-sentence ρ,

$$M{\left({\models }^{\mathrm{s}}\right)}_{\chi }^N\rho if and only ifN{\models }_{\square \chi }\rho .$$

Proposition 5.

For any signature morphism $(\phi ,\theta ):\chi \to {\chi }^{\prime }$ in ${\mathit{Sign}}^{\mathrm{s}}$, any ${\chi }^{\prime }$-model $M^{\prime }$, any $N^{\prime }\in {\left[\left[M^{\prime }\right]\right]}_{{\chi }^{\prime }}$, and any χ-sentence ρ:

$$M^{\prime }{\models }_{{\chi }^{\prime }}^{N^{\prime }}{\mathit{Sen}}^{\mathrm{s}}(\phi ,\theta )\rho if and only if{\mathit{Mod}}^{\mathrm{s}}(\phi ,\theta )M^{\prime }{\models }_{\chi }\rho .$$

Proof. 

By similarity to (9), we have that:

$$\mathit{Mod}\left(\phi \right)N^{\prime }\in {\left[\left[\mathit{Mod}\left(\theta \right)M^{\prime }\right]\right]}_{\chi }$$

(12)

Then, we have:

$$\begin{array}{ccc}\hfill M^{\prime }{\left({\models }^{\mathrm{s}}\right)}_{{\chi }^{\prime }}^{N^{\prime }}{\mathit{Sen}}^{\mathrm{s}}(\phi ,\theta )\rho & \iff N^{\prime }{\models }_{\square {\chi }^{\prime }}\mathit{Sen}\left(\phi \right)\rho \hfill & \hfill \mathrm{definition}\mathrm{of}{\models }^{\mathrm{s}}\\ & \iff \mathit{Mod}\left(\phi \right)N^{\prime }{\models }_{\square \chi }\rho \hfill & \hfill \mathrm{Satisfaction}\mathrm{Condition}\mathrm{of}\mathcal{I}\\ & \iff \mathit{Mod}\left(\theta \right)M^{\prime }{\left({\models }^{\mathrm{s}}\right)}_{\chi }^{\mathit{Mod}\left(\phi \right)N^{\prime }}\rho \hfill & \hfill \left(12\right),\mathrm{definition}\mathrm{of}{\models }^{\mathrm{s}}\\ & \iff \mathit{Mod}\left(\theta \right)M^{\prime }{\left({\models }^{\mathrm{s}}\right)}_{\chi }^{{\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}{N}^{\prime }}\hfill & \hfill \mathrm{definition}\mathrm{of}\left[\right[\_\left]\right]\\ & \iff {\mathit{Mod}}^{\mathrm{s}}(\phi ,\theta )M^{\prime }{\left({\models }^{\mathrm{s}}\right)}_{\chi }^{{\left[\left[M^{\prime }\right]\right]}_{(\phi ,\theta )}{N}^{\prime }}\hfill & \hfill \mathrm{definition}\mathrm{of}{\mathit{Mod}}^{\mathrm{s}}.\end{array}$$

□

By putting together propositions 1–5, we obtain:

Corollary 1.

${\mathcal{I}}^{\mathrm{s}}=({\mathit{Sign}}^{\mathrm{s}},{\mathit{Sen}}^{\mathrm{s}},{\mathit{Mod}}^{\mathrm{s}},\left[[\_]\right],{\models }^{\mathrm{s}})$ is a stratified institution.

The technical conditions underlying the construction of ${\mathcal{I}}^{\mathrm{s}}$ imposes some restriction both on the $\mathcal{I}$ signature morphisms $\chi$ that play the role of ${\mathit{Sign}}^{\mathrm{s}}$-signatures and on the $\mathcal{I}$ signature morphisms that make up the ${\mathit{Sign}}^{\mathrm{s}}$ morphisms. Let us see their significance and what they might mean in concrete situations.

• The $\mathcal{I}$ signature morphisms that stand as ${\mathit{Sign}}^{\mathrm{s}}$ signatures implicitly represent genuine partiality. By contrast, the $\mathcal{I}$ signature morphisms used for the ${\mathit{Sign}}^{\mathrm{s}}$ morphisms implicitly represent genuine totality, the reason being that we want to achieve totality for the syntactic and of the semantic translations at the level of the resultant stratified institution.
• The $\mathcal{I}$ signature morphisms standing as ${\mathit{Sign}}^{\mathrm{s}}$ signatures have to be fiber-small and quasi-representable. The former condition is necessary for the stratifications to be sets, and in the concrete situations is very mild. Only when we are in a many-sorted context doe it amount to a certain restriction, namely that there is no partiality of the translation of the sorts.
• In the applications, the quasi-representability condition on a signature morphism $\chi$ is less stringent than how it appears in principle.

  -

  As it is about (proper) model homomorphisms, it holds trivially in their absence. This degenerated situation is in fact the norm in the applications of the $3/2$-institutions, as until now there are not known applications that involve proper model homomorphisms.

  -

  When $\chi$ admits partiality only on the constants, then the quasi-representable holds.

  -

  When $\chi$ admits partiality only on the relation symbols and the model homomorphisms are “strong” (in the sense of [3]), then $\chi$ is quasi-representable, too.

• As an example that puts together some of the situations discussed above, if we consider the $3/2$-institution of many sorted first-order logics with “strong” model homomorphisms, then any signature morphism that is total on the sorts and on nonconstant operation symbols qualifies as a ${\mathit{Sign}}^{\mathrm{s}}$ signature.

4. Consequences of the Representation

4.1. Representing $3/2$-Institutions as Ordinary Institutions

In [6], a general representation of stratified institutions as ordinary institutions was developed. In [14], it is shown that this constitutes a left adjoint functor from the category $\mathbf{S}\mathbf{IN}\mathbf{S}$ of stratified institution morphisms to the category $\mathbf{IN}\mathbf{S}$ of ordinary institution morphisms. Let us recall this representation from either [6] or [14]. Given a stratified institution $\mathcal{S}=(\mathit{Sign},\mathit{Sen},\mathit{Mod},[[\_]],\models )$, the following institution ${\mathcal{S}}^{♯}=(\mathit{Sign},\mathit{Sen},{\mathit{Mod}}^{♯},{\models }^{♯})$ is defined by

• The objects of ${\mathit{Mod}}^{♯}(\Sigma )$ are the pairs $(M,w)$, such that $M\in \left|\mathit{Mod}\right(\Sigma \left)\right|$ and $w\in {\left[\left[M\right]\right]}_{\Sigma }$;
• The $\Sigma$-homomorphisms $(M,w)\to (N,v)$ are the pairs $(h,w)$, such that $h:M\to N$ and ${\left[\left[h\right]\right]}_{\Sigma }w=v$;
• For any signature morphism $\phi :\Sigma \to {\Sigma }^{\prime }$ and any ${\Sigma }^{\prime }$-model $(M^{\prime },w^{\prime })$

  $${\mathit{Mod}}^{♯}\left(\phi \right)(M^{\prime },w^{\prime })=(\mathit{Mod}\left(\phi \right)M^{\prime },{\left[\left[M^{\prime }\right]\right]}_{\phi }{w}^{\prime });$$
• For each $\Sigma$-model M, each $w\in {\left[\left[M\right]\right]}_{\Sigma }$, and each $\rho \in \mathit{Sen}(\Sigma )$

  $$\left((M,w){\models }_{\Sigma }^{♯}\rho \right)=\left(M{\models }_{\Sigma }^w\rho \right).$$

  (13)

By “composing” the representation of $3/2$-institutions as stratified institutions with the representation of stratified institutions as ordinary institutions, we obtain the following representation of $3/2$-institutions as ordinary institutions.

Corollary 2.

Let $\mathcal{I}=(\mathit{Sign},\mathit{Sen},\mathit{Mod},\models )$ be a $3/2$-institution. Then,

$${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}=({\mathit{Sign}}^{\mathrm{s}},{\mathit{Sen}}^{\mathrm{s}},{\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯},{\models }^{♯})$$

defines an ordinary institution where

• ${\mathit{Sign}}^{\mathrm{s}}$ and ${\mathit{Sen}}^{\mathrm{s}}$ are given by definitions 2 and 3, respectively.
• For each $\chi \in |{\mathit{Sign}}^{\mathrm{s}}|$:

  -

  A ${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}$χ-model is pair $(M,N)$ such that $M\in \left|\mathit{Mod}\right(\chi \square \left)\right|$, $N\in \mathit{Mod}\left(\chi \right)M$;

  -

  A χ-model homomorphism $(M,N)\to (M_0,N_0)$ is a model homomorphism $h:M\to M_0$, such that $N_0=h^N\square$.

• For each $(\phi ,\theta ):\chi \to {\chi }^{\prime }$ and any ${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}$${\chi }^{\prime }$-model $(M^{\prime },N^{\prime })$

  $${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}(\phi ,\theta )(M^{\prime },N^{\prime })=(\mathit{Mod}\left(\theta \right)M^{\prime },\mathit{Mod}\left(\phi \right)N^{\prime }).$$
• For each ${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}$χ-model $(M,N)$ and each ${\mathit{Sen}}^{\mathrm{s}}$ χ-sentence ρ

  $$(M,N){\models }_{\chi }^{♯}\rho if and only ifN{\models }_{\square \chi }^{\mathcal{I}}\rho .$$

4.2. Semantic Connectives

Institution theory has developed its own general approach to logical connectives [3,24,25]. This was refined in [6] to stratified institution theory. With $3/2$-institutions, there are two ways to approach this issue.

• The straightforward way that mimics the semantic treatment of connectives from ordinary institution theory.
• By using the stratified institution theoretic approach via the representation result given by corollary 1.

We argue that the straightforward approach does not work, which means that in order to have sound semantic connectives, we have to rely on the representation result. Our argument is based on an important property of the semantic connectives, namely that they should be preserved by the translations along signature morphisms. For instance, for a signature morphism $\phi$, if $\rho$ is a semantic disjunction of ${\rho }_1$ and ${\rho }_2$ in the signature $\square \phi$, then $\mathit{Sen}\left(\phi \right)\rho$ should be a semantic disjunction of $\mathit{Sen}\left(\phi \right){\rho }_1$ and $\mathit{Sen}\left(\phi \right){\rho }_2$ in $\phi \square$. This holds naturally in ordinary institution theory as well as in stratified institution theory, the proof of this relying on the satisfaction condition. In fact, in the stratified institutions case, this property can be established from the corresponding ordinary institution theory property via the ${\mathcal{S}}^{♯}$ representation, since as noticed in [6], the common propositional connectives and the quantification connectives do coincide in $\mathcal{S}$ and in ${\mathcal{S}}^{♯}$.

In order to understand what is wrong with the straightforward approach to the semantic connectives in $3/2$-institutions, let us attempt to establish the preservation property for the semantic disjunction. Let $\phi$ be a signature morphism and assume that $\rho$ is a semantic disjunction of ${\rho }_1$ and ${\rho }_2$ in $\square \phi$, which means that for each $\square \phi$-model M, $M\models \rho$ if and only if $M\models {\rho }_k$ for some $k\in \{1,2\}$. We have to establish the same property for $\mathit{Sen}\left(\phi \right)\rho$, $\mathit{Sen}\left(\phi \right){\rho }_1$, and $\mathit{Sen}\left(\phi \right){\rho }_2$. A first issue with this is the existence of these translations. We can overcome this by requiring that $\mathit{Sen}\left(\phi \right)\rho$ is a semantic disjunction of $\mathit{Sen}\left(\phi \right){\rho }_1$ and $\mathit{Sen}\left(\phi \right){\rho }_2$ when all three translations do exist. Let us attempt to prove the property under this new formulation. We have to prove that for any $\phi \square$-model $M^{\prime }$,

$$M\models \mathit{Sen}\left(\phi \right)\rho \mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f}{M}^{\prime }\models \mathit{Sen}\left(\phi \right){\rho }_{k}forsomek\in \{1,2\}.$$

However, $M^{\prime }\models \mathit{Sen}\left(\phi \right)\rho$ means $M\models \rho$ for all $M\in \mathit{Mod}\left(\phi \right)M^{\prime }$. Since $\rho$ is the semantic disjunction of ${\rho }_1$ and ${\rho }_2$, this further means that for each $M\in \mathit{Mod}\left(\phi \right)M^{\prime }$, there exists $k\in \{1,2\}$, such that $M\models {\rho }_k$. At this point, we have to get back to $\phi \square$, i.e., to establish that there exists $k\in \{1,2\}$ such that $M^{\prime }\models \mathit{Sen}\left(\phi \right){\rho }_k$, which means that for all $M\in \mathit{Mod}\left(\phi \right)M^{\prime }$, $M\models {\rho }_k$ for the samek. This is a gap because for one M we may have $M\models {\rho }_1$ and for another M we may have $M\models {\rho }_2$. So, the property cannot be established.

This failure to prove the preservation of semantic disjunctions along signature morphisms also tells us about the crucial role played by the reducts; that in fact the satisfaction in $3/2$-institutions has the reducts as an implicit parameter. This perspective provides a solution to our problem. Additionally, here we are; this situation calls for a stratified institution approach. In the particular case of the semantic disjunctions, this means that, under the representation given by corollary 1, we should define $\rho$ as a semantic disjunction of ${\rho }_1$ and ${\rho }_2$, when for each $\chi$-model M:

$$\{N\in {\left[\left[M\right]\right]}_{\chi }\mid N\models \rho \}=\{N\in {\left[\left[M\right]\right]}_{\chi }\mid N\models {\rho }_1\}\cup \{N\in {\left[\left[M\right]\right]}_{\chi }\mid N\models {\rho }_2\}.$$

Similar definitions can be derived in the case of the other semantic connectives by following the stratified institutions approach [6].

4.3. Model Amalgamation

Model amalgamation is one of the most important concepts/properties in institution theory. The institution theory literature contains numerous works where model amalgamation is used decisively. Refs. [2,26], etc., are representative for computing science-oriented works, especially in the area of software modularisation, while in [3] and many other articles, one may find an abundance of uses of model amalgamation in institution-independent model theory. Regarding its role in $3/2$-institution theory and applications, in [13], it is argued that model amalgamation squares in $3/2$-institutions constitute a superior approach to the categorical modeling of conceptual blending than $3/2$ or lax colimits.

The most notorious form of model amalgamation comes from ordinary institution theory. Given a diagram of signature morphisms, a model of that is a family ${\left(M_i\right)}_{i\in I}$ of models, indexed by the nodes of the diagram, such that $M_i$ is a ${\Sigma }_i$-model, where ${\Sigma }_i$ is the signature at node i, and such that for each signature morphism $\phi :{\Sigma }_i\to {\Sigma }_j$ in the diagram, $M_i=\mathit{Mod}\left(\phi \right)M_j$. A cocone $\mu$ of the diagram has the model amalgamation property when for each model ${\left(M_i\right)}_{i\in I}$ of the diagram there exists an unique model M of the vertex of the colimit, such that $\mathit{Mod}\left({\mu }_i\right)M=M_i$, $i\in I$. Then M is called the amalgamation of ${\left(M_i\right)}_{i\in I}$.

The most frequent use of model amalgamation is for cocones of spans of signature morphisms (which are in fact commutative squares of signature morphisms). There are also variations of the concept of model amalgamation: when we do not require the uniqueness of the amalgamation M (called weak model amalgamation), or when we refer only to colimits (called exactness) or even to particular colimits such as pushout squares (called semiexactness).

In stratified institution theory, there is a specific concept of model amalgamation called stratified model amalgamation, which corresponds to model amalgamation in the flattening ${\mathcal{S}}^{♯}$ of the respective stratified institution $\mathcal{S}$. This has been introduced in [14]. When the stratified institution is strict, stratified model amalgamation collapses to ordinary model amalgamation. Though in our context, this does not happen because the stratifications of the representations of $3/2$-institutions as stratified institutions are proper lax natural transformations.

The $3/2$-institution theoretic concept of model amalgamation [13] represents another refinement of the ordinary concept of model amalgamation. Its definition just replaces the ordinary definition of model amalgamation equalities relations with membership relations (for instance $M_i=\mathit{Mod}\left(\phi \right)M_j$ becomes $M_i\in \mathit{Mod}\left(\phi \right)M_j$) and strict commutativity with lax commutativity. $3/2$-institutional model amalgamation goes at the heart of the applications of $3/2$-institutions because the $3/2$-institution theoretic approach to conceptual blending comes with the proposal [13] to replace the original approach based on $3/2$-categorical colimits [10,11] with model amalgamation cocones.

The following result establishes an equivalence relationship between stratified model amalgamation in ${\mathcal{I}}^s$ and $3/2$-model amalgamation in $\mathcal{I}$.

Proposition 6.

Let $\mathcal{I}=(\mathit{Sign},\mathit{Sen},\mathit{Mod},\models )$ be a stratified institution and let ${\mathcal{I}}^{\mathrm{s}}=({\mathit{Sign}}^{\mathrm{s}},{\mathit{Sen}}^{\mathrm{s}},{\mathit{Mod}}^{\mathrm{s}},\left[[\_]\right],{\models }^{\mathrm{s}})$ be its representation as a stratified institution. Let the left hand square below represent a commutative diagram in ${\mathit{Sign}}^{\mathrm{s}}$, such that its projection on the first component (the right hand side square below) is a model amalgamation square in $\mathcal{I}$.

(14)

Then, the square of ${\mathcal{I}}^{\mathrm{s}}$-signature morphisms is a stratified model amalgamation square if and only if the following lax cocone of $\mathcal{I}$-signature morphisms has the model amalgamation property.

(15)

Proof. 

In this proof, we rely on the fact that stratified model amalgamation in ${\mathcal{I}}^{\mathrm{s}}$ is the same as model amalgamation in ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$. First, we show that the model amalgamation in $\mathcal{I}$ implies that in ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$.

In ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$, we consider a ${\chi }^{\prime }$-model $(M^{\prime },N^{\prime })$ and a ${\chi }_1$-model $(M_1,N_1)$, such that

$${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}(\zeta ,\eta )(M_1,N_1)=(M,N)and{\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}({\zeta }^{\prime },{\eta }^{\prime })(M^{\prime },N^{\prime })=(M,N).$$

(16)

Let the ${\Sigma }_1^{\prime }$-model $N_1^{\prime }$ be the unique amalgamation of $N^{\prime }$ and $N_1$. Then, $(M_1,M,M^{\prime },N_1,$$N,N^{\prime },N_1^{\prime })$ is a model for the ${\mathit{Sign}}^{\mathrm{s}}$ diagram with 7 vertices and 9 (full) arrows in (15). Let $M_1^{\prime }$ be the ${\Omega }_1^{\prime }$-model that is the unique amalgamation of $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$. We have that $(M_1^{\prime },N_1^{\prime })$ is a ${\chi }_1^{\prime }$-model and that:

$${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}({\phi }_1,{\theta }_1)(M_1^{\prime },N_1^{\prime })=(M_1,N_1)and{\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}({\zeta }^{\prime },{\eta }^{\prime })(M_1^{\prime },N_1^{\prime })=(M^{\prime },N^{\prime }).$$

The uniqueness of the amalgamation $(M_1^{\prime },N_1^{\prime })$ follows from the uniquenesses of the amalgamations $N_1^{\prime }$ and $M_1^{\prime }$.

Conversely, we now assume the amalgamation property in ${\mathcal{I}}^{\mathrm{s}}$ and prove it at the level of $\mathcal{I}$. Any model $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$ for the ${\mathit{Sign}}^{\mathrm{s}}$ diagram with 7 vertices and 9 arrows determines two ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$ models: an ${\chi }^{\prime }$-model $(M^{\prime },N^{\prime })$ and an ${\chi }_1$-model $(M_1,N_1)$, such that (16) holds. By the stratified model amalgamation property in ${\mathcal{I}}^{\mathrm{s}}$, interpreted in ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$, there exists an unique amalgamation $(M_1^{\prime },N_1^{″})$ of $(M^{\prime },N^{\prime })$ and of $(M_1,N_1)$. Since $\mathit{Mod}\left(\eta \right)N_1=\mathit{Mod}\left(\theta \right)N^{\prime }$, $\mathit{Mod}\left({\theta }_1\right)N_1^{\prime }=N_1$, $\mathit{Mod}\left({\eta }^{\prime }\right)N_1^{\prime }=N^{\prime }$, and because the same holds for $N_1^{″}$ in the place of $N_1^{\prime }$ by the uniqueness property of the $\mathcal{I}$-model amalgamation square (14), we have that $N_1^{″}=N_1^{\prime }$.

Now, we show that $M_1^{\prime }$ is the amalgamation of $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$.

• $N_1^{\prime }\in \mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }$ holds by the definition of ${\left({\mathit{Mod}}^{\mathrm{s}}\right)}^{♯}$ because $(M_1^{\prime },N_1^{\prime })$ is an ${\left({\mathcal{I}}^{\mathrm{s}}\right)}^{♯}$ ${\chi }_1^{\prime }$-model.
• Since $(M_1^{\prime },N_1^{\prime })$ is the amalgamation of $(M_1,N_1)$ and $(M^{\prime },N^{\prime })$, it follows that $\mathit{Mod}\left({\theta }_1\right)M_1^{\prime }=M_1$ and $\mathit{Mod}\left({\eta }^{\prime }\right)M_1^{\prime }=M^{\prime }$. It further follows that $\mathit{Mod}(\eta ;{\theta }_1)M_1^{\prime }=\mathit{Mod}(\theta ;{\eta }^{\prime })M_1^{\prime }=M$.
• We also have that:

  $$\begin{array}{ccc}\hfill N^{\prime }& =\mathit{Mod}\left({\zeta }^{\prime }\right)N_1^{\prime }\hfill & \hfill \mathrm{definition}\mathrm{of}{N}_1^{\prime }\\ & \in \mathit{Mod}\left({\zeta }^{\prime }\right)\left(\mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }\right)\hfill & \hfill N_1^{\prime }\in \mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }\\ & \subseteq \mathit{Mod}({\zeta }^{\prime };{\chi }_1^{\prime })M_1^{\prime }\hfill & \hfill \mathit{Mod}\mathrm{lax}.\end{array}$$
• By a similar argument to the above one, we establish that $N_1\in \mathit{Mod}({\phi }_1;{\chi }_1^{\prime })M_1^{\prime }$.
• Finally,

  $$\begin{array}{ccc}\hfill N& =\mathit{Mod}(\zeta ;{\phi }_1)N_1^{\prime }\hfill & \hfill N=\mathit{Mod}\left(\zeta \right)N_1,N_1=\mathit{Mod}\left({\phi }_1\right)N_1^{\prime }\\ & \in \mathit{Mod}(\zeta ;{\phi }_1)\left(\mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }\right)\hfill & \hfill N_1^{\prime }\in \mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }\\ & \subseteq \mathit{Mod}(\zeta ;{\phi }_1;{\chi }_1^{\prime })M_1^{\prime }\hfill & \hfill \mathit{Mod}\mathrm{lax}\\ & \subseteq \mathit{Mod}(\zeta ;{\chi }_1;{\theta }_1)M_1^{\prime }\hfill & \hfill {\chi }_1;{\theta }_1\le {\phi }_1;{\chi }_1^{\prime },\mathit{Mod}3/2-\mathrm{functor}\\ & \subseteq \mathit{Mod}(\chi ;\eta ;{\theta }_1)M_1^{\prime }\hfill & \hfill \chi ;\eta \le \zeta ;{\chi }_1,\mathit{Mod}3/2-\mathrm{functor}.\end{array}$$

The uniqueness of $M_1^{\prime }$ follows from the uniqueness of the amalgamation $(M_1^{\prime },N_1^{\prime })$ by relying on the first implication of the proposition. □

In the context of proposition 6, the following general result provides a sufficient condition for the lax cocone of $\mathcal{I}$-signature morphisms to have the model amalgamation property. Then, by the conclusion of proposition 6, this leads to the left-hand side square of diagrams (14) to be a stratified model amalgamation square. We need to recall from [13] two concepts as follows:

• In any $3/2$-category, a strict commutative square

  is a $3/2$-pushout when for any strict cocones $({\theta }_1^{\prime },{\theta }_2^{\prime })$ and $({\theta }_1^{″},{\theta }_2^{″})$ over the span $({\phi }_1,{\phi }_2)$, if ${\theta }_k^{\prime }\le {\theta }_k^{″}$, $k=\overline{1,2}$, there exists unique mediating arrows ${\mu }^{\prime }\le {\mu }^{″}$, such that ${\theta }_k;{\mu }^{\prime }={\theta }_k^{\prime }$ and ${\theta }_k^{″};{\mu }^{″}={\theta }_k^{″}$, $k=\overline{1,2}$. Note that $3/2$-pushouts are stronger than ordinary pushouts.
• $3/2$-institutional seeds were mentioned above when we discussed examples of $3/2$-institutions. For the full definition, see [13]. For the purpose of proposition 7 below, we only need the property that there exists a signature $\Pi$, such that for each signature $\Sigma$

  $$\left|\mathit{Mod}\right(\Sigma \left)\right|=\{M:\Sigma \to \Pi \mid \mathit{Sen}(M\left)\mathrm{total}\right\}$$

  and for each signature morphism $\phi$ and for each $\phi \square$-model $M^{\prime }$,

  $$\mathit{Mod}\left(\phi \right)M^{\prime }=\{M\in |\mathit{Mod}(\square \phi )|\mid \phi ;M^{\prime }\le M\}.$$

Proposition 7.

Let us assume a $3/2$-institution $\mathcal{I}$, such that when we remove its model homomorphisms it is generated by a $3/2$-institutional seed. Let us consider a lax cocone of $\mathcal{I}$ signature morphisms such as in diagrams (15) with the following properties:

• The inner square (also known as the right-hand side square of diagram (14)) is a $3/2$-pushout square;
• The outer square $(\eta ,\theta ,{\eta }^{\prime },{\theta }_1)$ is a model amalgamation square;
• $({\phi }_1,{\theta }_1):{\chi }_1\to {\chi }_1^{\prime }$ and $({\zeta }^{\prime },{\eta }^{\prime }):{\chi }^{\prime }\to {\chi }_1^{\prime }$ are strict.

Then, the lax cocone of diagram (15) has the model amalgamation property.

Proof. 

Let us consider $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$, a model of the diagram of 7 vertices and 9 arrows of diagrams (14). Let $M_1^{\prime }$ be the amalgamation of $M^{\prime }$ and $M_1$ by using the model amalgamation property of the outer square $(\eta ,\theta ,{\eta }^{\prime },{\theta }_1)$. We show that $M_1^{\prime }$ is also the amalgamation of $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$, and its uniqueness follows from the uniqueness as amalgamation of $M^{\prime }$ and $M_1$ only.

We first prove that $N_1^{\prime }\in \mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }$. This goes as follows. Since $N^{\prime }\in \mathit{Mod}\left({\chi }^{\prime }\right)M^{\prime }$, this means that ${\chi }^{\prime };M^{\prime }\le N^{\prime }$. It follows that:

$${\zeta }^{\prime };{\chi }_1^{\prime };M_1^{\prime }={\chi }^{\prime };{\eta }^{\prime };M_1^{\prime }={\chi }^{\prime };M^{\prime }\le N^{\prime }=\mathit{Mod}\left({\zeta }^{\prime }\right)N_1^{\prime }={\zeta }^{\prime };N_1^{\prime }.$$

(17)

Similarly,

$${\phi }_1;{\chi }_1^{\prime };M_1^{\prime }\le {\phi }_1;N_1^{\prime }.$$

(18)

Because the inner square of diagram (15), i.e., the square $(\phi ,\zeta ,{\zeta }^{\prime },{\phi }_1)$, is a $3/2$-pushout square, from (17) and (18), we obtain that ${\chi }_1^{\prime };M_1^{\prime }\le N_1^{\prime }$, which means $N_1^{\prime }\in \mathit{Mod}\left({\chi }_1^{\prime }\right)M_1^{\prime }$. From here, the proof that $M_1^{\prime }$ is an amalgamation of $(M_1,M,M^{\prime },N_1,N,N^{\prime },N_1^{\prime })$ follows the same steps as in the second part of the proof of proposition 6. In particular, this means that the three specific conditions of the current proposition are not used anymore. □

In concrete situations, it is quite common that the $3/2$-pushout condition on the inner square in proposition 7 implies the model amalgamation condition on the same square in proposition 6, a fact that enhances the applicability of proposition 7 within the context of the equivalence established by proposition 6.

Proposition 8.

In the context of a $3/2$-institution generated by a $3/2$-institutional seed, let us assume that $\mathit{Sen}$ preserves and reflects maximality (i.e., φ is maximal if and only if $\mathit{Sen}\left(\phi \right)$ is total). Then, any pushout cocone of signature morphisms determines a model amalgamation square.

Proof. 

Let $({\phi }_1,{\zeta }^{\prime })$ be a pushout cocone for a span $(\phi ,\zeta )$ of signature morphisms (such as in the diagram below).

Given a ${\Sigma }^{\prime }$-model $N^{\prime }$ and a ${\Sigma }_1$-model $N_1$, such that $\mathit{Mod}\left(\phi \right)N^{\prime }=\mathit{Mod}\left(\zeta \right)N_1$, we obtain a strict cocone $(N^{\prime },N_1)$ for the span $(\phi ,\zeta )$. By the pushout property of $({\phi }_1,{\zeta }^{\prime })$, there exists an unique $N_1^{\prime }:{\Sigma }_1^{\prime }\to \Pi$, such that $N^{\prime }={\zeta }^{\prime };N_1^{\prime }$ and $N_1={\phi }_1;N_1^{\prime }$. It remains to prove that $N_1^{\prime }$ qualifies as a ${\Sigma }_1^{\prime }$-model, i.e., that $\mathit{Sen}\left(N_1^{\prime }\right)$ is total.

By the hypothesis on $\mathit{Sen}$ that it preserves maximality, it is enough to prove that $N_1^{\prime }$ is maximal. Consider any $x:{\Sigma }_1^{\prime }\to \Pi$, such that $N_1^{\prime }\le x$. It follows that $N^{\prime }\le {\zeta }^{\prime };x$ and $N_1\le {\phi }_1;x$. By the hypothesis on $\mathit{Sen}$ that it reflects maximality, we have that both $N^{\prime }$ and $N_1$ are maximal, hence ${\zeta }^{\prime };x=N^{\prime }$ and ${\phi }_1;x=N_1$. By the pushout hypothesis, it follows that $x=N_1^{\prime }$. □

With respect to the conditions underlying proposition 8, note that:

• There are no restrictions on the signature morphisms that form the pushout square.
• Then, the condition of proposition 8 that $\mathit{Sen}$ preserves and reflects maximality applies well in concrete situations. As an example, let us consider the case of $3/2\mathcal{P}\mathcal{L}$. There, $\mathit{Sign}=\mathbf{P}\mathbf{f}\mathbf{n}$, and therefore it is evident that for any signature morphism $\phi$$\mathit{Sen}\left(\phi \right)$ is total if and only if $\phi$ is total.
• The $3/2$-pushout condition of proposition 7 in general is stronger than the pushout condition of proposition 8.

Often, in concrete situations that are related to the basic context of proposition 6, the pushout squares of signature morphisms are already $3/2$-pushout squares. The following result illustrates such a case that is emblematic for the concrete applications not only because it is sometimes involved as such (e.g., in $3/2\mathcal{P}\mathcal{L}$) but also because when it is not the case then the respective category of signature morphisms can be often treated in a similar way.

Proposition 9.

Any pushout square in $\mathbf{S}\mathbf{et}$ is a $3/2$-pushout square in $\mathbf{P}\mathbf{f}\mathbf{n}$.

Proof. 

For this proof, it is convenient to use the representation of partial functions as homomorphisms between pointed sets. A pointed set A is a set with a universally designated element ⊥. A homomorphism $f:A\to B$ of pointed sets is a function that preserves the designated element ⊥, i.e., $f\perp =\perp$. This yields a category $\mathbf{S}{\mathbf{et}}_{\perp }$ and a canonical isomorphism $\mathbf{P}\mathbf{f}\mathbf{n}\cong \mathbf{S}{\mathbf{et}}_{\perp }$ that:

• Maps any set A to the set $A_{\perp }=A\cup \{\perp \}$ (disjoint union);
• Maps any partial function $f:A\to B$ to the homomorphism $f_{\perp }:A_{\perp }\to B_{\perp }$ defined for each $x\in A$ by:

  $$f_{\perp }x=\left\{\begin{array}{cc}fx,\hfill & fx\mathrm{defined}\hfill \\ \perp ,\hfill & fx\mathrm{undefined}.\hfill \end{array}\right.$$

Now, let us consider a pushout square in $\mathbf{S}\mathbf{et}$ as follows.

(19)

By mapping the pushout square (19) to $\mathbf{S}{\mathbf{et}}_{\perp }$, we obtain the commutative square

This is a pushout square because ${(\_)}_{\perp }:\mathbf{S}\mathbf{et}\to \mathbf{S}{\mathbf{et}}_{\perp }$ is a left adjoint to the forgetful functor $\mathbf{S}{\mathbf{et}}_{\perp }\to \mathbf{S}\mathbf{et}$ and left adjoint functors preserve all colimits [15]. This left adjoint property can be either checked directly, or else, it can be established by noticing that it is a special case of a free algebra construction corresponding to a reduct functor of categories of algebras for the signature inclusion of the signature consisting of one sort into the signature consisting of one sort and one constant.

This showed that (19) is a pushout square in $\mathbf{P}\mathbf{f}\mathbf{n}$. In order to prove that this is a $3/2$-pushout square in $\mathbf{P}\mathbf{f}\mathbf{n}$, we let $(a^{\prime },a_1)$ and $(b^{\prime },b_1)$ be strict cocones for the span $(\phi ,\zeta )$, such that $a^{\prime }\subseteq b^{\prime }$ and $a_1\subseteq b_1$. Let $\alpha$ and $\beta$ be the unique mediating partial functions for $(a^{\prime },a_1)$ and $(b^{\prime },b_1)$, respectively. We have to show that $\alpha \subseteq \beta$. Note that $\alpha \subseteq \beta$ means that for each $x\in {\Sigma }_1^{\prime }$, ${\alpha }_{\perp }x\ne \perp$ implies ${\alpha }_{\perp }x={\beta }_{\perp }x$. However, ${\alpha }_{\perp }\ne \perp$ implies $x\ne \perp$. By the pushout property of (19), it follows that there exists y in ${\Sigma }^{\prime }$ or in ${\Sigma }_1$, such that ${\zeta }^{\prime }y=x$ or ${\phi }_{1}x=y$. By symmetry, without any loss of generality, we may assume that $y\in {\Sigma }^{\prime }$ and ${\zeta }^{\prime }y=x$. We have that:

$${\alpha }_{\perp }x={\alpha }_{\perp }\left({\zeta }^{\prime }y\right)=ay=by(\mathrm{s}\mathrm{i}\mathrm{n}\mathrm{c}\mathrm{e} a\subseteq b)={\beta }_{\perp }\left({\zeta }^{\prime }y\right)={\beta }_{\perp }x.$$

□

5. Conclusions and Future Work

We have defined a representation of $3/2$-institutions as stratified institutions in which the set of the reducts of a model with respect to a fixed signature morphism is assimilated to the set of its states. This representation is subject to some conditions on the signature morphisms. Then, we have explored three consequences of this general representation: a further representation to ordinary institutions, (stratified) semantic connectives in $3/2$-institutions, and stratified model amalgamation in $3/2$-institutions.

The results of our work also raise a series of issues to be addressed in the future. We mention a couple of them:

• The import of more model theory from stratified institution theory to $3/2$-institutions.
• Find general ways, with good applicability in concrete situations, to generate model amalgamation cocones such as in diagrams (15).

As stratified institution theory continues to develop, our representation result may provide new enhancements of the theory of $3/2$-institutions with concepts and results that come from stratified institutions.