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

Let A be an algebraic structure or equivalently 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. 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 φi 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 of integers under addition is a reduct of the group of integers under addition and negation, obtained by omitting negation. By contrast, the monoid of natural numbers under addition is not the reduct of any group.
Conversely the group is the expansion of the monoid, expanding it with the operation of negation.