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The primary concept in the design of the Magma system is that of
a `magma'. Following Bourbaki, a magma can be defined as
a set with a law of composition.
Thus, types correspond to magmas; a collection of magmas sharing a
common representation forms a category (e.g., the category of
permutation groups); a collection of categories satisfying the same
set of identical relations forms a variety (e.g., the variety of
groups). Functors may be used to move between categories, and the
variety operations (substructure, homomorphic image, and Cartesian
product) are available as uniform constructors across all categories.
While every value in Magma belongs to a unique parent magma, the
system provides a mechanism for automatic and forced coercion, to
move a value from one magma to another. To take a simple example,
when an integer is added to a rational, Magma automatically coerces
the integer into the rational field, so that the addition operation
can be carried out on two values with the same parent. Other
situations require forced coercion, in which the user states the
desired parent magma with the ! operator.
Example
Parents and categories:
> K := GF(7);
> P<x> := PolynomialRing(K);
> print Category(K), Category(P);
FldFin RngUPol
> e := 5;
> print Parent(e), Category(e);
Integer Ring
RngIntElt
Forced coercion:
> print Parent(K ! e);
Finite field of size 7
> print Parent(P ! e);
Univariate Polynomial Algebra in x over Finite field of size 7
Automatic coercion:
> f := x^2 + e + 6; print f;
x^2 + 4
> print Parent(f) eq P;
true
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