The Locally Free Relatively Filtered Diagram as an Inductive Completion of a System of Choice

Abstract

Guitart and Lair have established the existence of Locally Free Diagrams, which can be seen as a purely categorical version of the solution set condition, and of the Lowenheim–Skolem theorem. Their proof is based on a transfinite construction by saturation. An iterative principle is established, but the construction is not effective for every step. The thesis of Gerner contains a more effective proof for the existence of Locally Free Diagrams (with the restriction that the projective bases of the sketch S must all be finite). But the problem of lies in the impossibility to name concretely the elements of the Locally Free Diagrams. The present paper will provide a new construction of the Locally Free Diagram in which the effective and the non-effective part will be much more separated (again the projective bases must all be finite). This construction represents a notable improvement with regard to the proof of allowing the concrete designation of the elements ofthe Locally Free Diagrams. Furthermore we show that the construction is relatively filtered (i.e. satisfies the “filtered”-property).