<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% --> <!-- %% --> <!-- %A mapping.xml GAP documentation Thomas Breuer --> <!-- %% --> <!-- %% --> <!-- %Y (C) 1998 School Math and Comp. Sci., University of St Andrews, Scotland --> <!-- %Y Copyright (C) 2002 The GAP Group --> <!-- %% -->
<Chapter Label="Mappings">
<Heading>Mappings</Heading>
<Index Subkey="as in mathematics">functions</Index>
<Index>relations</Index>
A <E>mapping</E> in &GAP; is what is called a <Q>function</Q> in mathematics.
&GAP; also implements <E>generalized mappings</E> in which one element might
have several images, these can be imagined as subsets of the cartesian
product and are often called <Q>relations</Q>.
<P/>
Most operations are declared for general mappings and therefore this manual
often refers to <Q>(general) mappings</Q>, unless you deliberately need the
generalization you can ignore the <Q>general</Q> bit and just read
it as <Q>mappings</Q>.
<P/>
<#Include Label="[1]{mapping}">
<P/>
For mappings which preserve an algebraic structure a <E>kernel</E> is
defined.
Depending on the structure preserved the operation to compute this kernel is
called differently,
see Section <Ref Sect="Mappings which are Compatible with Algebraic Structures"/>.
<P/>
Some technical details of general mappings are described in
section <Ref Sect="General Mappings"/>.
<!-- %% The general support for mappings is due to Thomas Breuer. -->
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="sect:Direct Products and their Elements">
<Heading>Direct Products and their Elements</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Properties and Attributes of (General) Mappings">
<Heading>Properties and Attributes of (General) Mappings</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Images under Mappings">
<Heading>Images under Mappings</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Preimages under Mappings">
<Heading>Preimages under Mappings</Heading>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Arithmetic Operations for General Mappings">
<Heading>Arithmetic Operations for General Mappings</Heading>
<#Include Label="[3]{mapping}">
</Section>
<!-- %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% -->
<Section Label="Mappings which are Compatible with Algebraic Structures">
<Heading>Mappings which are Compatible with Algebraic Structures</Heading>
From an algebraical point of view, the most important mappings are those
which are compatible with a structure. For Magmas, Groups and Rings, &GAP;
supports the following four types of such mappings:
<P/>
<Enum>
<Item>
General mappings that respect multiplication
</Item>
<Item>
General mappings that respect addition
</Item>
<Item>
General mappings that respect scalar mult.
</Item>
<Item>
General mappings that respect multiplicative and additive structure
</Item>
</Enum>
<P/>
(Very technical note:
&GAP; defines categories <C>IsSPGeneralMapping</C> and
<C>IsNonSPGeneralMapping</C>.
The distinction between these is orthogonal to the structure compatibility
described here and should not be confused.)
Also see Sections <Ref Sect="Mappings that Respect Multiplication"/>,
<Ref Sect="Mappings that Respect Addition"/>,
and <Ref Attr="KernelOfMultiplicativeGeneralMapping"/>,
<Ref Attr="CoKernelOfMultiplicativeGeneralMapping"/>.
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