# Reduct

In universal algebra and in model theory, a **reduct** of an algebraic structure is obtained by omitting some of the operations and relations of that structure. The converse of "reduct" is "expansion."

## Definition[edit]

Let *A* be an algebraic structure (in the sense of universal algebra) or a structure in the sense of model theory, organized as a set *X* together with an indexed family of operations and relations φ_{i} on that set, with index set *I*. Then the **reduct** of *A* defined by a 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[edit]

The monoid (**Z**, +, 0) of integers under addition is a reduct of the group (**Z**, +, −, 0) of integers under addition and negation, obtained by omitting negation. By contrast, the monoid (**N**,+,0) of natural numbers 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[edit]

- Burris, Stanley N.; H. P. Sankappanavar (1981).
*A Course in Universal Algebra*. Springer. ISBN 3-540-90578-2. - Hodges, Wilfrid (1993).
*Model theory*. Cambridge University Press. ISBN 0-521-30442-3.