Constructor

The term "constructor" is used in Magma to mean an entity which returns some value, but is not a call to a function or operator.

Constructors of sets, sequences and tuples have special bracketing symbols around them. For instance, the indexed set {@ 6, 2, 8 @} is created by a constructor bracketed with the compound symbols {@ and @}, and the tuple <x^2 + 3, 4> (where x is an indeterminate of a polynomial ring, say) has the bracketing symbols < and > . More complex forms of set and sequence constructors have several parts within the bracketing. See Sets and Sequences.

However, most constructors are of the form NAME< LEFTSIDE | RIGHTSIDE > The NAME is frequently a short lower-case word, the most common of these being:

elt

element constructor

sub

submagma constructor

quo

quotient magma constructor

ncl

normal closure constructor

ideal

ideal constructor

lideal

left ideal constructor

rideal

right ideal constructor

ext

extension constructor

func

function constructor

case

case-expression constructor

map

mapping constructor

pmap

partial mapping constructor

hom

homomorphism constructor

The contents of the LEFTSIDE and RIGHTSIDE are highly dependent upon which constructor is being used and which category is involved. In general, apart from func , LEFTSIDE specifies the magma used for the construction, and RIGHTSIDE is a list of objects such as generators or relations, from which the constructor will build the new object, relative to LEFTSIDE.

A few constructors have names beginning with capital letters. These constructors are typically one-step versions of a two-step process: creating a generic magma, and forming a submagma or quotient. Examples are Semigroup and Monoid.

Some constructors can return more than one value, when a Multiple Assignment is used. The main value will always be the newly-constructed object, and the second value will be the associated mapping. For instance, quo returns the quotient magma as its principal value, and the natural homomorphism as its second value.

Example

> gaussians<i> := sub< MatrixRing(Integers(), 2) | [0, 1, -1, 0] >;
> gaussians;
Matrix Algebra of degree 2 with 1 generator over Integer Ring

> F<a, b> := FreeGroup(2);
> G<c, d> := quo<F | a^2, b^3, (a * b)^4 >;
> G;
Finitely presented group G on 2 generators
Relations
    c^2 = Id(G)
    d^3 = Id(G)
    (c * d)^4 = Id(G)
>
> // OR, IN ONE STEP: 
> G1<c, d> := Group< c, d | c^2, d^3, (c * d)^4 >;
> G1;                                       
Finitely presented group G1 on 2 generators
Relations
    c^2 = Id(G1)
    d^3 = Id(G1)
    (c * d)^4 = Id(G1)