---
tag: F4WRD396PB
kind: concept
name: Reduced product
msc: [08B25, 03C20]
sources:
  - url: https://en.wikipedia.org/wiki/Reduced_product
same_as:
  wikidata: Q7306336
  wikipedia: Reduced product
---

## Summary

A reduced product of [[algebras|QJGPT2W0FT]] is a quotient of their [[direct product|WJ73XYBNE4]].
It identifies two elements when they agree on a "large" set of indices.
A [[filter|15PF73BKY6]] on the index set says which sets count as large.
When the filter is an [[ultrafilter|J64Y3JC0RE]], the reduced product is an [[ultraproduct|AX9Q86GB8M]].
Reduced products preserve all identities and, more generally, all Horn sentences, which makes them useful in universal algebra and model theory.

## Explanations

### L1 human

Elements of a product are lists with one entry from each factor.
In a reduced product, two lists count as equal if they agree almost everywhere, where "almost everywhere" is fixed in advance by a filter.
For example, with the filter of cofinite subsets of $\N$, two sequences are equal when they differ in only finitely many places.
The operations are still computed entry by entry.