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Duality theorems for finite structures (characterising gaps and good characterisations). (English) Zbl 1024.05078
Summary: We provide a correspondence between the subjects of duality and density in classes of finite relational structures. The purpose of duality is to characterise the structures \(C\) that do not admit a homomorphism into a given target \(B\) by the existence of a homomorphism from a structure \(A\) into \(C\). Density is the order-theoretic property of containing no covers (or “gaps”). We show that the covers in the skeleton of a category of finite relational models correspond naturally to certain instances of duality statements, and we characterise these covers.

MSC:
05C75 Structural characterization of families of graphs
08A02 Relational systems, laws of composition
68R05 Combinatorics in computer science
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