In
universal algebra and in
model theory, a reduct of an
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set ...
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
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
''X'' together with an
indexed family of operations and relations φ
''i'' on that set, with
index set
In mathematics, an index set is a set whose members label (or index) members of another set. For instance, if the elements of a set may be ''indexed'' or ''labeled'' by means of the elements of a set , then is an index set. The indexing consis ...
''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
The
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids a ...
(Z, +, 0) of
integers under
addition
Addition (usually signified by the plus symbol ) is one of the four basic operations of arithmetic, the other three being subtraction, multiplication and division. The addition of two whole numbers results in the total amount or '' sum'' of ...
is a reduct of the
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
(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=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
Algebra
Mathematical relations
Model theory
Universal algebra