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.

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 φi on that set, with index set I.

That is, this reduct is the structure A with the omission of those operations and relations φi for which i is not in J.

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.