Ultrapower

Ultrapower

An ultrapower is an ultraproduct for which all the factors ${\displaystyle M_{i}}$ are equal. Explicitly, the ultrapower of a set ${\displaystyle M}$ modulo ${\displaystyle {\mathcal {U}}}$ is the ultraproduct ${\displaystyle {\textstyle \prod \limits _{i\in I}}M_{i}\,/\,{\mathcal {U}}={\textstyle \prod }_{\mathcal {U}}\,M_{\bullet }}$ of the indexed family ${\displaystyle M_{\bullet }:=\left(M_{i}\right)_{i\in I}}$ defined by ${\displaystyle M_{i}:=M}$ for every index ${\displaystyle i\in I.}$ The ultrapower may be denoted by ${\displaystyle {\textstyle \prod }_{\mathcal {U}}\,M}$ or (since ${\displaystyle {\textstyle \prod \limits _{i\in I}}M}$ is often denoted by ${\displaystyle M^{I}}$) by $${\displaystyle M^{I}/{\mathcal {U}}~:=~\prod _{i\in I}M\,/\,{\mathcal {U}}\,}$$