*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* I = (*Sign*<sup>I</sup> , *Sen*<sup>I</sup> , *Mod*<sup>I</sup> , |=<sup>I</sup> ) consists of


Such that for each morphism *ϕ* ∈ *Sign*<sup>I</sup> , the *satisfaction condition*

$$M' \stackrel{\mathcal{T}}{=}\_{\varrho \Box} \operatorname{Sen}^{\mathcal{T}}(\varrho)\rho \quad \text{if and only if} \quad M \stackrel{\mathcal{T}}{=}\_{\Box \varrho} \rho \tag{5}$$

holds for each *M* ∈ |*Mod*<sup>I</sup> (*ϕ*✷)|, each *M* ∈ |*Mod*<sup>I</sup> (*ϕ*)*M* |, and each *ρ* ∈ dom(*Sen*<sup>I</sup> (*ϕ*)).

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 *ϕ*, its corresponding sentence translation *Sen*(*ϕ*) is a partial function *Sen*(✷*ϕ*) → *Sen*(*ϕ*✷) and, moreover, whenever *ϕ* ≤ *θ*, we have that *Sen*(*ϕ*) ⊆ *Sen*(*θ*). The sentence functor *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 *Sen* is strict, while some general results require it to be either lax, oplax or strict.

The model reduct *Mod*(*ϕ*) is a lax functor *Mod*(*ϕ*✷) → P*Mod*(✷*ϕ*), implying that for each Σ -model *M* we have a *class of reducts M* rather than a single reduct. In concrete examples, this is a direct consequence of the partiality of *ϕ*: in the reducts, the interpretation of the symbols on which *ϕ* 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.


$$\operatorname{Mod}(\mathfrak{q})(\operatorname{Mod}(\mathfrak{q}')\operatorname{M}^{\prime\prime}) \subseteq \operatorname{Mod}(\mathfrak{q};\mathfrak{q}')\operatorname{M}^{\prime\prime}$$

and for each signature Σ and for each Σ-model *M* that

$$M \in \operatorname{Mod}(\mathbf{1}\_{\Sigma})M.$$

– The lax aspect of the reduct functors *Mod*(*ϕ*) means that for model homomorphisms *h*1, *h*2, such that *h*1✷ = ✷*h*2, we have that

$$\operatorname{Mod}(\mathfrak{q})(h\_1); \operatorname{Mod}(\mathfrak{q})(h\_2) \subseteq \operatorname{Mod}(\mathfrak{q})(h\_1; h\_2)$$

and for each *M* ∈ *Mod*(*ϕ*✷) and each *M* ∈ *Mod*(*ϕ*)*M* that

$$1\_M \in \operatorname{Mod}(\mathfrak{q}) 1\_{M'}.$$

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 *ϕ* in a 3/2-institution


$$\mathcal{Mod}(\varrho); \mathcal{Mod}(\theta) = \mathcal{Mod}(\theta; \varrho).$$

In general, in many concrete situations of interest, a signature morphism is *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.

