# Reduct

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Omission of operations and relations of a structure

This article is about a relation on algebraic structures. For reducts in abstract rewriting, see [Confluence (abstract rewriting)](/source/Confluence_(abstract_rewriting)).

In [universal algebra](/source/Universal_algebra) and in [model theory](/source/Model_theory), a **reduct** of an [algebraic structure](/source/Algebraic_structure) is obtained by omitting some of the [operations](/source/Operation_(mathematics)) and [relations](/source/Relation_(mathematics)) of that structure. The opposite of "reduct" is "expansion".

## Definition

Let *A* be an algebraic structure (in the sense of [universal algebra](/source/Universal_algebra)) or a [structure](/source/Structure_(mathematical_logic)) in the sense of [model theory](/source/Model_theory), organized as a [set](/source/Set_(mathematics)) *X* together with an [indexed family](/source/Indexed_family) of operations and relations φ*i* on that set, with [index set](/source/Index_set) *I*. Then the **reduct** of *A* defined by a [subset](/source/Subset) *J* of *I* is the structure consisting of the set *X* and *J*-indexed family of operations and relations whose *j*-th operation or relation for *j* ∈ *J* is the *j*-th operation or relation of *A*. That is, this reduct is the structure *A* with the omission of those operations and relations φ*i* for which *i* is not in *J*.

A structure *A* is an **expansion** of *B* just when *B* is a reduct of *A*. That is, reduct and expansion are mutual converses.

## Examples

The [monoid](/source/Monoid) (**Z**, +, 0) of [integers](/source/Integer) under [addition](/source/Addition) is a reduct of the [group](/source/Group_(mathematics)) (**Z**, +, −, 0) of integers under addition and [negation](/source/Additive_inverse), obtained by omitting negation. By contrast, the monoid (**N**, +, 0) of [natural numbers](/source/Natural_number) under addition is not the reduct of any group.

Conversely the group (**Z**, +, −, 0) is the expansion of the monoid (**Z**, +, 0), expanding it with the operation of negation.

## References

- Burris, Stanley N.; H. P. Sankappanavar (1981). [*A Course in Universal Algebra*](http://www.thoralf.uwaterloo.ca/htdocs/ualg.html). [Springer](/source/Springer_Science%2BBusiness_Media). [ISBN](/source/ISBN_(identifier)) [3-540-90578-2](https://en.wikipedia.org/wiki/Special:BookSources/3-540-90578-2).

- Hodges, Wilfrid (1993). [*Model theory*](https://archive.org/details/modeltheory0000hodg). [Cambridge University Press](/source/Cambridge_University_Press). [ISBN](/source/ISBN_(identifier)) [0-521-30442-3](https://en.wikipedia.org/wiki/Special:BookSources/0-521-30442-3).

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Adapted from the Wikipedia article [Reduct](https://en.wikipedia.org/wiki/Reduct) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Reduct?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.