Examples of parents endowed with multiple realizations

sage.categories.examples.with_realizations.SubsetAlgebra

An example of parent endowed with several realizations

We consider an algebra \(A(S)\) whose bases are indexed by the subsets \(s\) of a given set \(S\). We consider three natural basis of this algebra: F, In, and Out. In the first basis, the product is given by the union of the indexing sets. That is, for any \(s, t\subset S\)

\[F_s F_t = F_{s\cup t}\]

The In basis and Out basis are defined respectively by:

\[In_s = \sum_{t\subset s} F_t \qquad\text{and}\qquad F_s = \sum_{t\supset s} Out_t\]

Each such basis gives a realization of \(A\), where the elements are represented by their expansion in this basis.

This parent, and its code, demonstrate how to implement this algebra and its three realizations, with coercions and mixed arithmetic between them.

EXAMPLES:

sage: A = Sets().WithRealizations().example(); A
The subset algebra of {1, 2, 3} over Rational Field
sage: A.base_ring()
Rational Field

The three bases of A:

sage: F   = A.F()  ; F
The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
sage: In  = A.In() ; In
The subset algebra of {1, 2, 3} over Rational Field in the In basis
sage: Out = A.Out(); Out
The subset algebra of {1, 2, 3} over Rational Field in the Out basis

One can quickly define all the bases using the following shortcut:

sage: A.inject_shorthands()
Defining F as shorthand for The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
Defining In as shorthand for The subset algebra of {1, 2, 3} over Rational Field in the In basis
Defining Out as shorthand for The subset algebra of {1, 2, 3} over Rational Field in the Out basis

Accessing the basis elements is done with basis() method:

sage: F.basis().list()
[F[{}], F[{1}], F[{2}], F[{3}], F[{1, 2}], F[{1, 3}], F[{2, 3}], F[{1, 2, 3}]]

To access a particular basis element, you can use the from_set() method:

sage: F.from_set(2,3)
F[{2, 3}]
sage: In.from_set(1,3)
In[{1, 3}]

or as a convenient shorthand, one can use the following notation:

sage: F[2,3]
F[{2, 3}]
sage: In[1,3]
In[{1, 3}]

Some conversions:

sage: F(In[2,3])
F[{}] + F[{2}] + F[{3}] + F[{2, 3}]
sage: In(F[2,3])
In[{}] - In[{2}] - In[{3}] + In[{2, 3}]

sage: Out(F[3])
Out[{3}] + Out[{1, 3}] + Out[{2, 3}] + Out[{1, 2, 3}]
sage: F(Out[3])
F[{3}] - F[{1, 3}] - F[{2, 3}] + F[{1, 2, 3}]

sage: Out(In[2,3])
Out[{}] + Out[{1}] + 2*Out[{2}] + 2*Out[{3}] + 2*Out[{1, 2}] + 2*Out[{1, 3}] + 4*Out[{2, 3}] + 4*Out[{1, 2, 3}]

We can now mix expressions:

sage: (1 + Out[1]) * In[2,3]
Out[{}] + 2*Out[{1}] + 2*Out[{2}] + 2*Out[{3}] + 2*Out[{1, 2}] + 2*Out[{1, 3}] + 4*Out[{2, 3}] + 4*Out[{1, 2, 3}]