Distributive substructural logics as coalgebraic logics over posets

0379885 - ÚI 2013 RIV GB eng C - Conference Paper (international conference)
Bílková, Marta - Horčík, Rostislav - Velebil, J.
Distributive substructural logics as coalgebraic logics over posets.
Advances in Modal Logic. London: College Publications, 2012 - (Bolander, T.; Braüner, T.; Moss, L.; Ghilardi, S.), s. 119-142. 9. ISBN 978-1-84890-068-4.
[Advances in Modal Logic. Copenhagen (DK), 22.08.2012-25.08.2012]
R&D Projects: GA ČR GAP202/11/1632
Institutional research plan: CEZ:AV0Z10300504
Institutional support: RVO:67985807
Keywords : substructural logics * frame semantics * coalgebras * coalgebraic logic * Goldblatt-Thomason theorem
Subject RIV: BA - General Mathematics

We show how to understand frame semantics of distributive substructural logics coalgebraically, thus opening a possibility to study them as coalgebraic logics. As an application of this approach we prove a general version of Goldblatt-Thomason theorem that characterizes definability of classes of frames for logics extending the distributive Full Lambek logic, as e.g. relevance logics, many-valued logics or intuitionistic logic. The paper is rather conceptual and does not claim to contain significant new results. We consider a category of frames as posets equipped with monotone relations, and show that they can be understood as coalgebras for an endofunctor of the category of posets. In fact, we adopt a more general definition of frames that allows to cover a wider class of distributive modal logics. Goldblatt-Thomason theorem for classes of resulting coalgebras for instance shows that frames for axiomatic extensions of distributive Full Lambek logic are modally definable classes of certain coalgebras, the respective modal algebras being precisely the corresponding subvarieties of distributive residuated lattices.
Permanent Link: http://hdl.handle.net/11104/0210742