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

0506928 - ÚI 2020 RIV NL eng J - Journal Article
Galatos, N. - Gil-Férez, José
Modules over quantaloids: Applications to the isomorphism problem in algebraic logic and pi-institutions.
Journal of Pure and Applied Algebra. Roč. 221, č. 1 (2017), s. 1-24. ISSN 0022-4049. E-ISSN 1873-1376
R&D Projects: GA ČR GAP202/10/1826
Institutional support: RVO:67985807
Keywords : abstract algebraic logic * isomorphism of logics * pi-institutions * algebra
OECD category: Pure mathematics
Impact factor: 0.720, year: 2017
Method of publishing: Limited access
http://dx.doi.org/10.1016/j.jpaa.2016.05.012

We solve the isomorphism problem in the context of abstract algebraic logic and of π-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 π-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 π-institutions.
Permanent Link: http://hdl.handle.net/11104/0298058