A logic of implications in algebra and coalgebra

• DOI: 10.1007/s00012-009-0009-z
• Publication Date: 2009-08-28T12:57:05Z
• Authors: Jiří Adámek; Manuela Sobral; Lurdes Sousa
• Keywords: Finitary; Coalgebra; Epimorphism

Implications in a category can be presented as epimorphisms: an object satisfies the implication iff it is injective with respect to that epimorphism. G. Rosu formulated a logic for deriving an implication from other implications. We present two versions of implicational logics: a general one and a finitary one (for epimorphisms with finitely presentable domains and codomains). In categories Alg Σ of algebras on a given signature our logic specializes to the implicational logic of R. Quackenbush. In categories Coalg H of coalgebras for a given accessible endofunctor H of sets we derive a logic for implications in the sense of P. Gumm.