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The general method for getting ultraproducts uses an index set I, a structure $M_i$ for each element i of I (all of the same signature), and an ultrafilter U on I. The usual choice is for I to be infinite and U to contain all cofinite subsets of I. Otherwise the ultrafilter is principal, and the ultraproduct is isomorphic to one of the factors. (ultraproduct, Wikipedia)

What does "factor" refer to?

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up vote 5 down vote accepted

The answer to your question is in the six lines that immediately follow the sentence that you quoted. You start with an indexed family $\{M_i:i\in I\}$ of structures of the same type (linear orders, rings, fields, etc. $-$ technically, they have the same signature). You form the Cartesian product $$M=\prod_{i\in I}M_i\;;$$ the structures $M_i$ are the factors of this product. You take an ultrafilter $\mathscr{U}$ on $I$, and you use it to define an equivalence relation $\sim$ on $M$:

$$\langle m_i:i\in I\rangle\sim\langle m'_i:i\in I\rangle\text{ iff }\{i\in I:m_i=m'_i\}\in\mathscr{U}\;.$$ The ultraproduct of the $M_i$ is the quotient $M/\sim$, often written $M/\mathscr{U}$, whose elements are the $\sim$-equivalence clases. Although $M/\sim$ is no longer strictly speaking a product, we still speak of the $M_i$ $-$ the genuine factors of the Cartesian product $M$ $-$ as its factors.

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