Cartesian Product Constructor and Functions

The special constructor car< ... > is used for the creation of cartesian products of structures.

car< R₁, ..., Rₖ >: Str, ..., Str -> SetCart

Given a list of sets or algebraic structures \(R_1, \ldots, R_k\), construct the cartesian product set \(R_1 \times \cdots \times R_k\).

CartesianProduct(R, S): Str, ..., Str -> SetCart

Given structures \(R\) and \(S\), construct the cartesian product set \(R \times S\). This is the same as calling the car constructor with the two arguments \(R\) and \(S\).

CartesianProduct(L): [Str] -> SetCart

CartesianProduct(L): <Str> -> SetCart

Given a sequence or tuple \(L\) of structures, construct the cartesian product of the elements of \(L\).

CartesianPower(R, k): Str, RngIntElt -> SetCart

Given a structure \(R\) and an integer \(k\), construct the cartesian power set \(R^{k}\).

Flat(C): SetCart -> SetCart

Given a cartesian product \(C\) of structures which may themselves be cartesian products, return the cartesian product of the base structures, considered in depth-first order (see Flat for the element version).

NumberOfComponents(C): SetCart -> RngIntElt

Given a cartesian product \(C\), return the number of components of \(C\).

Component(C, i): SetCart, RngIntElt -> Str

C[i]: SetCart, RngIntElt -> Str

The \(i\)-th component of \(C\).

Components(C): SetCart -> List

The list of components of a cartesian product.

# C: SetCart -> RngIntElt

Given a cartesian product \(C\), return the cardinality of \(C\).

Rep(C): SetCart -> Elt

Given a cartesian product \(C\), return a representative of \(C\).

Random(C): SetCart -> Elt

Given a cartesian product \(C\), return a random element of \(C\).

Example: Cartesian Product of Q and Z (ex-c2f16a)

We create the product of \({\mathbb{Q}}\) and \({\mathbb{Z}}\).

> C := car< RationalField(), Integers() >;
> C;
Cartesian Product<Rational Field, Ring of Integers>

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