The Category Assertions part of your domain constructor definition lists those categories of which all domains created by the constructor are unconditionally members. The word ``unconditionally'' means that membership in a category does not depend on the values of the parameters to the domain constructor. This part thus defines the link between the domains and the category hierarchies given on the inside covers of this book. As described in ugCategoriesCorrectness it is this link that makes it possible for you to pass objects of the domains as arguments to other operations in Axiom.
Every QuadraticForm domain is declared to be unconditionally a member of category AbelianGroup. An abelian group is a collection of elements closed under addition. Every object x of an abelian group has an additive inverse y such that . The exports of an abelian group include , +, -, and scalar multiplication by an integer. After asserting that QuadraticForm domains are abelian groups, it is possible to pass quadratic forms to algorithms that only assume arguments to have these abelian group properties.
In ugCategoriesConditionals you saw that Fraction(R), a member of QuotientFieldCategory(R), is a member of OrderedSet if is a member of OrderedSet. Likewise, from the Exports part of the definition of ModMonic(R, S),
you see that ModMonic(R, S) is a member of Finite if is.
The Exports part of a domain definition is the same kind of expression that can appear to the right of an == in a category definition. If a domain constructor is unconditionally a member of two or more categories, a form is used. Join The Exports part of the definition of FlexibleArray(S) reads, for example: