Reduct
Encyclopedia
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."
(in the sense of universal algebra
) 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.
Structure A is an expansion of B just when B is a reduct of A. That is, reduct and expansion are mutual converses.
(Z, +, 0) of integer
s under addition
is a reduct of the group
(Z, +, −, 0) of integers under addition and negation, obtained by omitting negation.
Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
and in model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, 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 structureAlgebraic structure
In abstract algebra, an algebraic structure consists of one or more sets, called underlying sets or carriers or sorts, closed under one or more operations, satisfying some axioms. Abstract algebra is primarily the study of algebraic structures and their properties...
(in the sense of universal algebra
Universal algebra
Universal algebra is the field of mathematics that studies algebraic structures themselves, not examples of algebraic structures....
) or equivalently a structure
Structure (mathematical logic)
In universal algebra and in model theory, a structure consists of a set along with a collection of finitary operations and relations which are defined on it....
in the sense of model theory
Model theory
In mathematics, model theory is the study of mathematical structures using tools from mathematical logic....
, organized as a set X together with an indexed family
Family
In human context, a family is a group of people affiliated by consanguinity, affinity, or co-residence. In most societies it is the principal institution for the socialization of children...
of operations
Operation (mathematics)
The general operation as explained on this page should not be confused with the more specific operators on vector spaces. For a notion in elementary mathematics, see arithmetic operation....
and relations φi on that set, with index set
Index set
In mathematics, the elements of a set A may be indexed or labeled by means of a set J that is on that account called an 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.
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 monoidMonoid
In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single associative binary operation and an identity element. Monoids are studied in semigroup theory as they are naturally semigroups with identity. Monoids occur in several branches of mathematics; for...
(Z, +, 0) of integer
Integer
The integers are formed by the natural numbers together with the negatives of the non-zero natural numbers .They are known as Positive and Negative Integers respectively...
s under addition
Addition
Addition is a mathematical operation that represents combining collections of objects together into a larger collection. It is signified by the plus sign . For example, in the picture on the right, there are 3 + 2 apples—meaning three apples and two other apples—which is the same as five apples....
is a reduct of the group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
(Z, +, −, 0) of integers under addition and negation, obtained by omitting negation.
Conversely the group (Z, +, −, 0) is the expansion of the monoid (Z, +, 0), expanding it with the operation of negation.