Epimorphisms, Definability and Cardinalities

0503853 - ÚI 2021 RIV NL eng J - Journal Article
Moraschini, Tommaso - Raftery, J.G. - Wannenburg, J. J.
Epimorphisms, Definability and Cardinalities.
Studia Logica. Roč. 108, č. 2 (2020), s. 255-275. ISSN 0039-3215. E-ISSN 1572-8730
R&D Projects: GA ČR GA17-04630S
EU Projects: European Commission(XE) 689176 - SYSMICS
Institutional support: RVO:67985807
Keywords : Epimorphism * Prevariety * Quasivariety * Beth definability * Algebraizable logic * Equivalential logic
OECD category: Pure mathematics
Impact factor: 0.585, year: 2020
Method of publishing: Limited access
http://dx.doi.org/10.1007/s11225-019-09846-5

We characterize, in syntactic terms, the ranges of epimorphisms in an arbitrary class of similar first-order structures (as opposed to an elementary class). This allows us to strengthen a result of Bacsich, as follows: in any prevariety having at most s nonlogical symbols and an axiomatization requiring at most m variables, if the epimorphisms into structures with at most m + s + ℵ0 elements are surjective, then so are all of the epimorphisms. Using these facts, we formulate and prove manageable 'bridge theorems', matching the surjectivity of all epimorphisms in the algebraic counterpart of a logic ⊢ with suitable infinitary definability properties of ⊢ , while not making the standard but awkward assumption that ⊢ comes furnished with a proper class of variables.
Permanent Link: http://hdl.handle.net/11104/0295628