Constructing Ultrapowers from Elementary Extensions of Full Clones

TL;DR

The paper presents a method to represent algebras in the variety generated by a full clone algebra as limit reduced powers and ultrapowers, generalizing ultrapower constructions.

Contribution

It introduces a novel approach to represent algebras and ultrapowers as limit constructions, extending the ultrapower framework to broader algebraic contexts.

Findings

Representation of any algebra in the variety generated by a full clone as limit reduced powers.

Representation of Boolean ultrapowers as limit ultrapowers.

Extension of ultrapower constructions to general algebraic structures.

Abstract

Let $A$ be an infinite set. Let $\Omega(A)$ be the algebra over $A$ where every constant is a fundamental constant and every finitary function is a fundamental operation. We shall give a method of representing any algebra $\mathcal{L}$ in the variety generated by $\Omega(A)$ as limit reduced powers and even direct limits of limit reduced powers of $\mathcal{L}$. If the algebra $\mathcal{L}$ is elementarily equivalent to $\Omega(A)$, then this construction represents $\Omega(A)$ as a limit ultrapower and also as direct limits of limit ultrapowers of $\Omega(A)$. This method therefore gives a method of representing Boolean ultrapowers and other generalizations of the ultrapower construction as limit ultrapowers and direct limits of limit ultrapowers.