IsTotal( map ) P
is true if each element in the source S of the general mapping map
has images, i.e.,
smap ¹ Æ for all s Î S,
and false otherwise.
IsSingleValued( map ) P
is true if each element in the source S of the general mapping map
has at most one image, i.e.,
|smap | £ 1 for all s Î S,
and false otherwise.
Equivalently, IsSingleValued( map ) is true if and only if
the preimages of different elements in R are disjoint.
IsMapping( map ) P
A mapping map is a general mapping that assigns to each element elm
of its source a unique element Image( map, elm ) of its range.
Equivalently, the general mapping map is a mapping if and only if it is total and single-valued (see IsTotal, IsSingleValued).
IsInjective( map ) P
is true if the images of different elements in the source S of the
general mapping map are disjoint, i.e.,
xmap Çymap = Æ for x ¹ y Î S,
and false otherwise.
Equivalently, IsInjective( map ) is true if and only if each
element in the range of map has at most one preimage in S.
IsSurjective( map ) P
is true if each element in the range R of the general mapping map
has preimages in the source S of map, i.e.,
{ s Î S; x Î smap } ¹ Æ for all x Î R,
and false otherwise.
IsBijective( map ) P
A general mapping map is bijective if and only if it is an injective and surjective mapping (see IsMapping, IsInjective, IsSurjective).
Range( map ) A
Source( map ) A
UnderlyingRelation( map ) A
The underlying relation of a general mapping map is the domain
of pairs (s,r), with s in the source and r in the range of map
(see Source, Range), and r Î ImagesElm( map, s ).
Each element of the underlying relation is a tuple (see IsTuple).
UnderlyingGeneralMapping( map ) A
attribute for underlying relations of general mappings
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GAP 4 manual