Varieties of positive modal algebras and structural completeness

0504824 - ÚI 2020 RIV GB eng J - Journal Article
Moraschini, Tommaso
Varieties of positive modal algebras and structural completeness.
Review of Symbolic Logic. Roč. 12, č. 3 (2019), s. 557-588. ISSN 1755-0203. E-ISSN 1755-0211
R&D Projects: GA ČR(CZ) GF15-34650L; GA MŠMT(CZ) EF17_050/0008361
Institutional support: RVO:67985807
Keywords : positive modal logic * modal logic * structural completeness * admissible rule * abstract algebraic logic * algebraization of Gentzen systems
OECD category: Pure mathematics
Impact factor: 0.750, year: 2019
Method of publishing: Limited access
http://dx.doi.org/10.1017/S1755020319000236

Positive modal algebras are the〈∧,∨, 3, D, 0, 1〉-subreducts of modal algebras. We show that the variety of positive interior algebras is not locally finite. However, the free one-generated positive interior algebra has 37 elements. Moreover, we show that there are exactly 16 varieties of height at most 4 in the lattice of varieties of positive interior algebras. Building on this, we infer that there are only 3 non-trivial structurally complete varieties of positive K4-algebras. These are also the unique non-trivial hereditarily structurally complete such varieties. Moreover, we characterize passively structurally complete varieties of positive K4-algebras and show that there are infinitely many of them. These results are related to the study of structurally complete axiomatic extensions of an algebraizable Gentzen system for positive modal logic.
Permanent Link: http://hdl.handle.net/11104/0296383