An Algebraic Preservation Theorem for Aleph-Zero Categorical Quantified Constraint Satisfaction

TL;DR

This paper establishes an algebraic preservation theorem for positive Horn definability in countably categorical structures, introducing the concept of periodic power and periomorphisms, with applications to complexity classification of constraint satisfaction problems.

Contribution

It introduces the periodic power construction and periomorphisms, providing a new algebraic characterization of positive Horn definability in aleph-zero categorical structures.

Findings

Proves an equivalence between positive Horn definability and preservation by periomorphisms.

Provides a new proof for the complexity classification of quantified constraint satisfaction on equality templates.

Develops the concept of periodic power for algebraic analysis of structures.

Abstract

We prove an algebraic preservation theorem for positive Horn definability in aleph-zero categorical structures. In particular, we define and study a construction which we call the periodic power of a structure, and define a periomorphism of a structure to be a homomorphism from the periodic power of the structure to the structure itself. Our preservation theorem states that, over an aleph-zero categorical structure, a relation is positive Horn definable if and only if it is preserved by all periomorphisms of the structure. We give applications of this theorem, including a new proof of the known complexity classification of quantified constraint satisfaction on equality templates.