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arXiv · 1503.02189

Splitting methods in algebraic logic: Proving results on non-atom-canonicity, non-finite axiomatizability and non-first oder definability for cylindric and relation algebras

Tarek Sayed Ahmed
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Abstract

We deal with various splitting methods in algebraic logic. The word `splitting' refers to splitting some of the atoms in a given relation or cylindric algebra each into one or more subatoms obtaining a bigger algebra, where the number of subatoms obtained after splitting is adjusted for a certain combinatorial purpose. This number (of subatoms) can be an infinite cardinal. The idea originates with Leon Henkin. Splitting methods existing in a scattered form in the literature, possibly under different names, proved useful in obtaining (negative) results on non-atom canonicity, non-finite axiomatizability and non-first order definability for various classes of relation and cylindric algebras. In a unified framework, we give several known and new examples of each. Our framework covers Monk's splitting, Andr\'eka's splitting, and, also, so-called blow up and blur constructions involving splitting (atoms) in finite Monk-like algebras and rainbow algebras.

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Tarek Sayed Ahmed. 2015-03-07. Splitting methods in algebraic logic: Proving results on non-atom-canonicity, non-finite axiomatizability and non-first oder definability for cylindric and relation algebras. https://arxiv.org/abs/1503.02189

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