Section12.11

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## 12.11 Conditionals

As categories have parameters, the actual operations exported by a conditional category can depend on these parameters. As an example, the operation determinantdeterminantMatrixCategory from category MatrixCategory is only exported when the underlying domain R has commutative multiplication:

if R has commutative("*") then
determinant: $ -> R

Conditionals can also define conditional extensions of a category. Here is a portion of the definition of QuotientFieldCategory: QuotientFieldCategory

QuotientFieldCategory(R) : Category == ... with ...
if R has OrderedSet then OrderedSet
if R has IntegerNumberSystem then
ceiling: $ -> R
...

Think of category QuotientFieldCategory(R) as denoting the domain Fraction(R), the class of all fractions of the form a/b for elements of R. The first conditional means in English: ``If the elements of R are totally ordered ( R is an OrderedSet), then so are the fractions a/b''. Fraction

The second conditional is used to conditionally export an operation ceiling which returns the smallest integer greater than or equal to its argument. Clearly, ``ceiling'' makes sense for integers but not for polynomials and other algebraic structures. Because of this conditional, the domain Fraction(Integer) exports an operation ceiling: Fraction Integer->Integer, but Fraction Polynomial Integer does not.

Conditionals can also appear in the default definitions for the operations of a category. For example, a default definition for ceilingceilingField within the part following the add reads:

if R has IntegerNumberSystem then
ceiling x == ...

Here the predicate used is identical to the predicate in the Exports part. This need not be the case. See ugPackagesConds for a more complicated example.

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