A coalgebraic study of BL-algebras

Abstract

This thesis is intended to develop the theory of coalgebras over an endofunctor by investigating coalgebraic structure of BL-algebras via two functors. The first assigns every BL-algebra to its MV-center, and the second assigns every local BL-algebra to its quotient by its unique maximal filter. Functorial coalgebras have been mainly studied on the category of sets, topological spaces and also in arbitrary categories. Our aim is to prove that the categories of logical algebras are also good candidates as base categories of coalgebras. We study some categorical properties of BL-algebras and show that BL-algebras have good properties enough to obtain a rich structure of coalgebra on them ((co)completeness, adequate factorization structure, coreflective subcategory) . We introduce the MV-functor, and investigate its coalgebras. We characterize homomophisms , subcoalgebras, bisimulations and prove that the category of coalgebras of the MV-functor is complete and cocomplete. Moreover, we add a topological structure based on filters of the underlined BL-algebras and obtain topological MV-coalgebras. We construct an inverse system in the category of MV-coalgebras and show that the category of topological MV-coalgebras is complete, cocomplete and strong-monotopological over the category of MV-coalgebras. We also introduce Q-coalgebras over local BL-algebras ( which are BL-algebras with a unique maximal filter) and local BL-frames. We show that the corresponding categories are isomorphic, establishing a link between coalgebras over BL-algebras and modal logic