A filter is a special unary GAP function that returns either true
or false, depending on whether or not the argument lies in the set defined
by the filter.
Filters are used to express different aspects of information about a GAP
object, which are described below
(see Categories, Representation, Attributes,
Setter and Tester for Attributes, Properties, Other Filters).
Presently any filter in GAP is implemented as a function which
corresponds to a set of positions in the bitlist which forms part of the
type of each GAP object, and returns true if and only if the bitlist
of the type of the argument has the value true at all of these positions.
The intersection (or meet) of two filters filt1, filt2 is again a filter, it can be formed as
and filt2
For example, IsList and IsEmpty is a filter that returns true if
its argument is an empty list, and false otherwise. The filter
IsGroup is defined as the intersection of the category
IsMagmaWithInverses and the property IsAssociative.
A filter that is not the meet of other filters is called a simple filter. For example, each attribute tester (see Setter and Tester for Attributes) is a simple filter. Each simple filter corresponds to a position in the bitlist currently used as part of the data structure representing a type.
Every filter filt has a rank, which is used to define a ranking of the methods installed for an operation, see Section Method Installation in ``Programming in GAP''. The rank of a filter can be accessed as
RankFilter( filt ) F
For simple filters, an incremental rank is defined when the filter is created, see the sections about the creation of filters Creating Categories, Creating Representations, Creating Attributes and Properties, Creating Other Filters; all in ``Programming in GAP''. For an arbitrary filter, its rank is given by the sum of the incremental ranks of the involved simple filters; in addition to the implied filters, these are also the required filters of attributes (again see the sections about the creation of filters). In other words, for the purpose of computing the rank and only for this purpose, attribute testers are treated as if they would imply the requirements of their attributes.
NamesFilter( filt ) F
NamesFilter returns a list of names of the implied simple filters
of the filter filt, these are all those simple filters imp such that
every object in filt also lies in imp.
For implications between filters, see ShowImpliedFilters as well as
sections Logical Implications, Creating Categories,
Creating Representations, Creating Attributes and Properties in
``Programming in GAP''
ShowImpliedFilters( filter ) F
Displays information about the filters that may be implied by
filter. They are given by their names. ShowImpliedFilters first
displays the names of all filters that are unconditionally implied by
filter. It then displays implications that require further filters to
be present (indicating by + the required further filters).
The function displays only first-level implications, implications that
follow in turn are not displayed (though GAP will do these).
gap> ShowImpliedFilters(IsRationalFunction); Implies: IsNearAdditiveElementWithInverse IsAdditiveElement IsMultiplicativeElementWithInverse May imply with: +IsZero +Tester(IsZero) IsConstantRationalFunction Tester(IsConstantRationalFunction)
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