# Free constructions in algebraic institutions

## Abstract

To provide a formal framework for discussing specifications of algebraic abstract data types we introduce the notion of an algebraic institution. Our main results concern the problem of the existence of free constructions in algebraic institutions. We review a characterization of logical specification systems that guarantee the existence of initial models for any consistent set of axioms given by Mahr and Makowsky in [MM 83a, MM 83b]. Then the more general problem of the existence of free functors (left adjoints to forgetful functors) for any theory morphism is analysed. We give a construction of a free model of a theory over a model of a subtheory (with respect to an arbitrary theory morphism) which requires only the existence of initial models.

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## 8 References

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