Hierarchies of direct powers, ultrapowers and cumulative powers

TL;DR

This paper explores cumulative hierarchies of functions on structures, connecting them with direct powers and ultrapowers, and applies the framework to construct Conway's surreal field.

Contribution

It provides a new characterization of the preservation fragment of first-order theories and links cumulative powers with direct and ultrapowers.

Findings

Cumulative powers extend preservation phenomena of reduced, direct, and ultrapowers.

Direct powers and ultrapowers can be derived from cumulative powers via quotients.

Conditions are identified under which ultrapowers embed into cumulative or direct powers.

Abstract

In this paper we investigate cumulative hierarchies of functions on structures, or cumulative powers, and study their properties. Particularly, we show how they extend the preservation phenomena of reduced powers, direct powers and ultrapowers by offering a characterization of the fragment of first-order theory it preserves, and elucidate the connections between the three sorts of constructions. More precisely, we show how both direct powers and ultrapowers may be obtained from cumulative powers as quotients by appropriate equivalence relations. We address how embeddability lifts from generating structures to their cumulative powers, direct powers and ultrapowers, and under what conditions ultrapowers embed into corresponding cumulative powers or direct powers. We further offer an application of the framework to show a straightforward way of constructing Conway's surreal field.