Reduct
{{Short description|Omission of operations and relations of a structure}}
{{about|a relation on algebraic structures|reducts in abstract rewriting|Confluence (abstract rewriting)}}
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 opposite of "reduct" is "expansion".
Definition
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. 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 (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
- {{cite book | last=Burris | first=Stanley N. |author2=H. P. Sankappanavar | publisher = Springer | title=A Course in Universal Algebra | year=1981 | isbn=3-540-90578-2 | url=http://www.thoralf.uwaterloo.ca/htdocs/ualg.html }}
- {{cite book | last=Hodges | first=Wilfrid | publisher=Cambridge University Press | title=Model theory | url=https://archive.org/details/modeltheory0000hodg | url-access=registration | year=1993 | isbn=0-521-30442-3 }}
{{Mathematical logic}}