Examples of parents endowed with multiple realizations#
- class sage.categories.examples.with_realizations.SubsetAlgebra(R, S)#
Bases:
UniqueRepresentation
,Parent
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
, andOut
. 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 andOut
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.
See also
the Implementing Algebraic Structures thematic tutorial.
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}]
- class Bases(parent_with_realization)#
Bases:
Category_realization_of_parent
The category of the realizations of the subset algebra
- class ParentMethods#
Bases:
object
- from_set(*args)#
Construct the monomial indexed by the set containing the elements passed as arguments.
EXAMPLES:
sage: In = Sets().WithRealizations().example().In(); In The subset algebra of {1, 2, 3} over Rational Field in the In basis sage: In.from_set(2,3) In[{2, 3}]
As a shorthand, one can construct elements using the following notation:
sage: In[2,3] In[{2, 3}]
- one()#
Returns the unit of this algebra.
This default implementation takes the unit in the fundamental basis, and coerces it in
self
.EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: In = A.In(); Out = A.Out() sage: In.one() In[{}] sage: Out.one() Out[{}] + Out[{1}] + Out[{2}] + Out[{3}] + Out[{1, 2}] + Out[{1, 3}] + Out[{2, 3}] + Out[{1, 2, 3}]
- super_categories()#
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: C = A.Bases(); C Category of bases of The subset algebra of {1, 2, 3} over Rational Field sage: C.super_categories() [Category of realizations of The subset algebra of {1, 2, 3} over Rational Field, Join of Category of algebras with basis over Rational Field and Category of commutative algebras over Rational Field and Category of realizations of unital magmas]
- F#
alias of
Fundamental
- class Fundamental(A)#
Bases:
CombinatorialFreeModule
,BindableClass
The Subset algebra, in the fundamental basis
INPUT:
A
– a parent with realization inSubsetAlgebra
EXAMPLES:
sage: A = Sets().WithRealizations().example() sage: A.F() The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis sage: A.Fundamental() The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
- one()#
Return the multiplicative unit element.
EXAMPLES:
sage: A = AlgebrasWithBasis(QQ).example() # optional - sage.combinat sage.modules sage: A.one_basis() # optional - sage.combinat sage.modules word: sage: A.one() # optional - sage.combinat sage.modules B[word: ]
- one_basis()#
Returns the index of the basis element which is equal to ‘1’.
EXAMPLES:
sage: F = Sets().WithRealizations().example().F(); F The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis sage: F.one_basis() {} sage: F.one() F[{}]
- product_on_basis(left, right)#
Product of basis elements, as per
AlgebrasWithBasis.ParentMethods.product_on_basis()
.INPUT:
left
,right
– sets indexing basis elements
EXAMPLES:
sage: F = Sets().WithRealizations().example().F(); F The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis sage: S = F.basis().keys(); S Subsets of {1, 2, 3} sage: F.product_on_basis(S([]), S([])) F[{}] sage: F.product_on_basis(S({1}), S({3})) F[{1, 3}] sage: F.product_on_basis(S({1,2}), S({2,3})) F[{1, 2, 3}]
- class In(A)#
Bases:
CombinatorialFreeModule
,BindableClass
The Subset Algebra, in the
In
basisINPUT:
A
– a parent with realization inSubsetAlgebra
EXAMPLES:
sage: A = Sets().WithRealizations().example() sage: A.In() The subset algebra of {1, 2, 3} over Rational Field in the In basis
- class Out(A)#
Bases:
CombinatorialFreeModule
,BindableClass
The Subset Algebra, in the \(Out\) basis
INPUT:
A
– a parent with realization inSubsetAlgebra
EXAMPLES:
sage: A = Sets().WithRealizations().example() sage: A.Out() The subset algebra of {1, 2, 3} over Rational Field in the Out basis
- a_realization()#
Returns the default realization of
self
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.a_realization() The subset algebra of {1, 2, 3} over Rational Field in the Fundamental basis
- base_set()#
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.base_set() {1, 2, 3}
- indices()#
The objects that index the basis elements of this algebra.
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.indices() Subsets of {1, 2, 3}
- indices_key(x)#
A key function on a set which gives a linear extension of the inclusion order.
INPUT:
x
– set
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: sorted(A.indices(), key=A.indices_key) [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]
- supsets(set)#
Returns all the subsets of \(S\) containing
set
INPUT:
set
– a subset of the base set \(S\) ofself
EXAMPLES:
sage: A = Sets().WithRealizations().example(); A The subset algebra of {1, 2, 3} over Rational Field sage: A.supsets(Set((2,))) [{1, 2, 3}, {2, 3}, {1, 2}, {2}]