A mapping in GAP is what is called a ``function'' in mathematics. GAP also implements generalized mappings in which one element might have several images, these can be imagined as subsets of the cartesian product and are often called ``relations''.
Most operations are declared for general mappings and therefore this manual often refers to ``(general) mappings'', unless you deliberatily need the generalization you can ignore the ``general'' bit and just read it as ``mappings''.
A general mapping F in GAP is described by
its source S, its range R, and a subset Rel of the direct product
S ×R, which is called the underlying relation of F.
S, R, and Rel are generalized domains (see Chapter Domains).
The corresponding attributes for general mappings are Source, Range,
and UnderlyingRelation.
Note that general mappings themselves are not domains. One reason for this is that two general mappings with same underlying relation are regarded as equal only if also the sources are equal and the ranges are equal. Other, more technical, reasons are that general mappings and domains have different basic operations, and that general mappings are arithmetic objects (see Arithmetic Operations for General Mappings); both should not apply to domains.
Each element of an underlying relation of a general mapping lies in the category of tuples (see IsTuple).
For each s Î S, the set { r Î R | (s,r) Î Rel } is called the set of images of s. Analogously, for r Î R, the set { s Î S | (s,r) Î Rel } is called the set of preimages of r.
The ordering of general mappings via < is defined by the ordering
of source, range, and underlying relation. Specifically, if the Source
and Range domains of map1 and map2 are the same, then one considers
the union of the preimages of map1 and map2 as a strictly ordered set.
The smaller of map1 and map2 is the one whose image is smaller on the
first point of this sequence where they differ.
For mappings which preserve an algebraic structure a kernel is defined. Depending on the structure preserved the operation to compute this kernel is called differently, see section Mappings which are Compatible with Algebraic Structures.
Some technical details of general mappings are described in section General Mappings.
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GAP 4 manual