Constructing Ultrapowers from Elementary Extensions of Full Clones

Joseph Van Name

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arXiv · 1207.0118

Constructing Ultrapowers from Elementary Extensions of Full Clones

Joseph Van Name

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Abstract

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.

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BibTeX RIS

Joseph Van Name. 2012-06-30. Constructing Ultrapowers from Elementary Extensions of Full Clones. https://arxiv.org/abs/1207.0118

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