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7.2.2 map (plural)
Maps are ring maps from a preimage ring (source) into the basering (target), defined by specifying images for source variables in the target ring.
Note:
- the target of a map is ALWAYS the actual basering
- the preimage ring has to be stored "by its name", that means, maps can only be used in such contexts, where the name of the preimage ring can be resolved (this has to be considered in subprocedures). See also Identifier resolution, Names in procedures.
Maps between rings with different coefficient fields are possible and listed below.
Canonically realized are
-
(
: the rational numbers)
-
(
: the real numbers)
-
(
: the complex numbers)
-
(
: the integers)
-
(
: the Galois field)
-
-
Possible are furthermore
-
$Z/p \rightarrow Q,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2]
\subseteq Z$
-->
![$Z/p \rightarrow Q,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2]
\subseteq Z$](sing_83.png)
-
![$Z/p \rightarrow Z/p^\prime,
\quad
[i]_p \mapsto i \in [-p/2, \, p/2] \subseteq Z, \;
i \mapsto [i]_{p^\prime} \in Z/p^\prime$](sing_84.png)
-
by taking the real part
Finally, in PLURAL we allow the mapping from rings with coefficient field Q to rings whose ground fields have finite characteristic:
Note:
In these cases the denominator and the numerator
of a number are mapped separately by the usual
map from Z to Z/p, and the image of the number
is built again afterwards by division. It is thus
not allowed to map numbers whose denominator is
divisible by the characteristic of the target
ground field, or objects containing such numbers.
We, therefore, strongly recommend to use such
maps only to map objects with integer coefficients.
Note that - in contrast to the commutative case - maps between non-commutative rings
easily fail to be a morphism.
7.2.2.1 map declarations (plural) 7.2.2.2 map expressions (plural) 7.2.2.3 map (plural) operations 7.2.2.4 map related functions (plural)