The family of an object determines its relationship to other objects.
More precisely, the families form a partition of all GAP objects such that the following two conditions hold: objects that are equal w.r.t. ``='' lie in the same family; and the family of the result of an operation depends only on the families of its operands.
The first condition means that a family can be regarded as a set of elements instead of a set of objects. Note that this does not hold for categories and representations (see below), two objects that are equal w.r.t. ``='' need not lie in the same categories and representations. For example, a sparsely represented matrix can be equal to a densely represented matrix. Similarly, each domain is equal w.r.t. ``='' to the sorted list of its elements, but a domain is not a list, and a list is not a domain.
FamilyObj( obj ) F
returns the family of the object obj.
The family of the object obj is itself an object, its family is the
FamilyOfFamilies.
It should be emphasized that families may be created when they are needed. For example, the family of elements of a finitely presented group is created only after the presentation has been constructed. Thus families are the dynamic part of the type system, that is, the part that is not fixed after the initialisation of GAP.
Families can be parametrized. For example, the elements of each finitely presented group form a family of their own. Here the family of elements and the finitely presented group coincide when viewed as sets. Note that elements in different finitely presented groups lie in different families. This distinction allows GAP to forbid multiplications of elements in different finitely presented groups.
As a special case, families can be parametrized by other families. An important example is the family of collections that can be formed for each family. A collection consists of objects that lie in the same family, it is either a nonempty dense list of objects from the same family or a domain.
Note that every domain is a collection, that is, it is not possible to
construct domains whose elements lie in different families. For
example, a polynomial ring over the rationals cannot contain the
integer 0 because the family that contains the integers does not
contain polynomials. So one has to distinguish the integer zero from
each zero polynomial.
Let us look at this example from a different viewpoint. A polynomial ring and its coefficients ring lie in different families, hence the coefficients ring cannot be embedded ``naturally'' into the polynomial ring in the sense that it is a subset. But it is possible to allow, e.g., the multiplication of an integer and a polynomial over the integers. The relation between the arguments, namely that one is a coefficient and the other a polynomial, can be detected from the relation of their families. Moreover, this analysis is easier than in a situation where the rationals would lie in one family together with all polynomials over the rationals, because then the relation of families would not distinguish the multiplication of two polynomials, the multiplication of two coefficients, and the multiplication of a coefficient with a polynomial. So the wish to describe relations between elements can be taken as a motivation for the introduction of families.
GAP 4 manual