42.19 Mapping Records

A mapping map is represented by a record with the following components

isGeneralMapping:

always true, indicating that this is a general mapping.

source:

the source of the mapping, i.e., the set of elements to which the mapping can be applied.

range:

the range of the mapping, i.e., a set in which all value of the mapping lie.

The following entries are optional. The functions with the corresponding names will generally test if they are present. If they are then their value is simply returned. Otherwise the functions will perform the computation and add those fields to those record for the next time.

isMapping:

true if map is a single valued mapping and false otherwise.

isInjective
isSurjective
isBijection
isHomomorphism
isMonomorphism
isEpimorphism
isIsomorphism
isEndomorphism

isAutomorphism:

true if map has the corresponding property and false otherwise.

preImage:

the subset of map.source of elements pre that are actually mapped to at least one element, i.e., for which Images( map, pre ) is nonempty.

image:

the subset of map.range of the elements img that are actually values of the mapping, i.e., for which PreImages( map, img ) is nonempty.

inverseMapping:

the inverse mapping of map. This is a mapping from map.range to map.source that maps each element img to the set PreImages( map, img ).

The following entry is optional. It must be bound only if the inverse of map is indeed a single valued mapping.

inverseFunction:

the inverse function of map.

The following entry is optional. It must be bound only if map is a homomorphism.

kernel:

the elements of map.source that are mapped to the identity element of map.range.

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