Extension Properties and Subdirect Representation in Abstract Algebraic Logic

0484922 - ÚTIA 2019 RIV NL eng J - Journal Article
Lávička, Tomáš - Noguera, Carles
Extension Properties and Subdirect Representation in Abstract Algebraic Logic.
Studia Logica. Roč. 106, č. 6 (2018), s. 1065-1095. ISSN 0039-3215. E-ISSN 1572-8730
R&D Projects: GA ČR GA17-04630S
Institutional support: RVO:67985556
Keywords : Abstract algebraic logic * Infinitary logics * Natural extensions * Natural expansions * Semilinear logics * Subdirect representation
OECD category: Pure mathematics
Impact factor: 0.467, year: 2018
http://library.utia.cas.cz/separaty/2018/MTR/lavicka-0484922.pdf

This paper continues the investigation, started in Lávička and Noguera (Stud Log 105(3): 521–551, 2017), of infinitary propositional logics from the perspective of their algebraic completeness and filter extension properties in abstract algebraic logic. If follows from the Lindenbaum Lemma used in standard proofs of algebraic completeness that, in every finitary logic, (completely) intersection-prime theories form a basis of the closure system of all theories. In this article we consider the open problem of whether these properties can be transferred to lattices of filters over arbitrary algebras of the logic. We show that in general the answer is negative, obtaining a richer hierarchy of pairwise different classes of infinitary logics that we separate with natural examples. As by-products we obtain a characterization of subdirect representation for arbitrary logics, develop a fruitful new notion of natural expansion, and contribute to the understanding of semilinear logics.
Permanent Link: http://hdl.handle.net/11104/0280148