Constructing Ultrapowers from Elementary Extensions of Full Clones
Joseph Van Name · arXiv (Cornell University) · 2012
Let $A$ be an infinite set. Let $Ω(A)$ 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 $Ω(A)$ as limit reduced powers and even direct limits of limit reduced powers of $\mathcal{L}$. If the algebra $\mathcal{L}$ is elementarily equivalent to $Ω(A)$, then this construction represents $Ω(A)$ as a limit ultrapower and also as direct limits of limit ultrapowers of $Ω(A)$. 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.