Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and pi-institutions

作者：Galatos Nikolaos^*; Gil Ferez Jose

来源：Journal of Pure and Applied Algebra, 2017, 221(1): 1-24.

DOI：10.1016/j.jpaa.2016.05.012

We solve the isomorphism problem in the context of abstract algebraic logic and of pi-institutions, namely the problem of when the notions of syntactic and semantic equivalence among logics coincide. The problem is solved in the general setting of categories of modules over quantaloids. We introduce closure operators on modules over quantaloids and their associated morphisms. We show that, up to isomorphism, epis are morphisms associated with closure operators. The notions of (semi-)interpretability and (semi-)representability are introduced and studied. We introduce cyclic modules, and provide a characterization for cyclic projective modules as those having a g-variable. Finally, we explain how every pi-institution induces a module over a quantaloid, and thus the theory of modules over quantaloids can be considered as an abstraction of the theory of pi-institutions.

• 出版日期2017-1