Characterizing amalgmation bases for relation, cylindric and polyadic algebras

TL;DR

This paper provides a complete characterization of amalgamation bases for relation, cylindric, and polyadic algebras using special neat embeddings, and explores how expanding these algebras affects their amalgamation properties.

Contribution

It offers necessary and sufficient conditions for an algebra to be in the amalgamation bases of various classes, including infinite dimensions, and shows how natural expansions influence these properties.

Findings

Characterization of amalgamation bases for relation, cylindric, and polyadic algebras.

Expansion of algebras with natural operations affects their amalgamation properties.

Applicable to all dimensions, including infinite, with conditions based on neat embeddings.

Abstract

We characterize completey (give a necessary and suffcient condition using special neat embeddings)for a relation algebra to belong to the amalgamation, strong amalgamation, and superamalgamation base of the class of representable algebras. We do the same for cylindric and polyadic algebras for all dimensions >1, infinite included. Finally, we show that we can expand our algebras by finitely many natural operations, that force the newly expanded algebras to have various forms of amalgamation properties, like quasi-projections and directed cylindrifiers.