Blow up and Blur constructions in Algebraic Logic

TL;DR

This paper explores blow up and blur constructions in algebraic logic, demonstrating how finite structures can be expanded with infinite components to produce weakly but not strongly representable atom structures, revealing non-elementarity.

Contribution

It provides a rigorous framework for blow up and blur constructions, unifying various existing methods and illustrating the non-elementarity of strongly representable atom structures.

Findings

Weakly representable atom structures are not strongly representable.

Construction of graphs with finite and infinite blurs illustrates non-elementarity.

The framework unifies existing constructions in algebraic logic.

Abstract

The idea in the title is to blow up a finite structure, replacing each 'colour or atom' by infinitely many, using blurs to represent the resulting term algebra, but the blurs are not enough to blur the structure of the finite structure in the complex algebra. Then, the latter cannot be representable due to a {finite- infinite} contradiction. This structure can be a finite clique in a graph or a finite relation algebra or a finite cylindric algebra. This theme gives examples of weakly representable atom structures that are not strongly representable. Many constructions existing in the literature are placed in a rigorous way in such a framework, properly defined. This is the essence too of construction of Monk like-algebras, one constructs graphs with finite colouring (finitely many blurs), converging to one with infinitely many, so that the original algebra is also blurred at the complex algebra level, and the term algebra is completey representable, yielding a representation of its completion the complex algebra. A reverse of this process exists in the literature, it builds algebras with infinite blurs converging to one with finite blurs. This idea due to Hirsch and Hodkinson, uses probabilistic methods of Erdos to construct a sequence of graphs with infinite chromatic number one that is 2 colourable. This construction, which works for both relation and cylindric algebras, further shows that the class of strongly representable atom structures is not elementary.