*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 [6–8], etc.

**Citation:** Diaconescu, R. Representing 3/2-Institutions as Stratified Institutions. *Mathematics* **2022**, *10*, 1507. https://doi.org/ 10.3390/math10091507

Academic Editor: Gabriel Ciobanu

Received: 26 February 2022 Accepted: 28 April 2022 Published: 1 May 2022

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