Tutorial: Implementing Algebraic Structures

Author: Nicolas M. Thiéry <nthiery at users.sf.net>, Jason Bandlow <jbandlow@gmail.com> et al.

This tutorial will cover four concepts:

• endowing free modules and vector spaces with additional algebraic structure

• defining morphisms

• defining coercions and conversions

• implementing algebraic structures with several realizations

At the end of this tutorial, the reader should be able to reimplement by himself the example of algebra with several realizations:

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

Namely, we consider an algebra \(A(S)\) whose basis is indexed by the subsets \(s\) of a given set \(S\). \(A(S)\) is endowed with three natural basis: F, In, Out; in the first basis, the product is given by the union of the indexing sets. The In basis and Out basis are defined respectively by:

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

Each such basis gives a realization of \(A\), where the elements are represented by their expansion in this basis. In the running exercises we will progressively implement this algebra and its three realizations, with coercions and mixed arithmetic between them.

This tutorial heavily depends on Tutorial: Using Free Modules and Vector Spaces. You may also want to read the less specialized thematic tutorial How to implement new algebraic structures.

Subclassing free modules and including category information

As a warm-up, we implement the group algebra of the additive group \(\ZZ/5\ZZ\). Of course this is solely for pedagogical purposes; group algebras are already implemented (see ZMod(5).algebra(ZZ)). Recall that a fully functional \(\ZZ\)-module over this group can be created with the simple command:

sage: A = CombinatorialFreeModule(ZZ, Zmod(5), prefix='a')

We reproduce the same, but by deriving a subclass of CombinatorialFreeModule:

sage: class MyCyclicGroupModule(CombinatorialFreeModule):
....:     """An absolutely minimal implementation of a module whose basis is a cyclic group"""
....:     def __init__(self, R, n, *args, **kwargs):
....:         CombinatorialFreeModule.__init__(self, R, Zmod(n), *args, **kwargs)

sage: A = MyCyclicGroupModule(QQ, 6, prefix='a') # or 4 or 5 or 11     ...
sage: a = A.basis()
sage: A.an_element()
2*a[0] + 2*a[1] + 3*a[2]

We now want to endow \(A\) with its natural product structure, to get the desired group algebra. To define a multiplication, we should be in a category where multiplication makes sense, which is not yet the case:

sage: A.category()
Category of finite dimensional vector spaces with basis over Rational Field

We can look at the available Categories from the documentation in the reference manual or we can use introspection to look through the list of categories to pick one we want:

sage: sage.categories.<tab>                   # not tested

Once we have chosen an appropriate category (here AlgebrasWithBasis), one can look at one example:

sage: E = AlgebrasWithBasis(QQ).example(); E
An example of an algebra with basis: the free algebra on the generators ('a', 'b', 'c') over Rational Field
sage: e = E.an_element(); e
B[word: ] + 2*B[word: a] + 3*B[word: b] + B[word: bab]

and browse through its code:

sage: E??                                     # not tested

This code is meant as a template for implementing a new algebra. In particular, this template suggests that we need to implement the methods product_on_basis, one_basis, _repr_ and algebra_generators. Another way to get this list of methods is to ask the category (TODO: find a slicker idiom for this):

sage: from sage.misc.abstract_method import abstract_methods_of_class
sage: abstract_methods_of_class(AlgebrasWithBasis(QQ).element_class) # py2
{'optional': ['_add_', '_mul_'],
 'required': ['__nonzero__', 'monomial_coefficients']}
sage: abstract_methods_of_class(AlgebrasWithBasis(QQ).element_class) # py3
{'optional': ['_add_', '_mul_'],
 'required': ['__bool__', 'monomial_coefficients']}
sage: abstract_methods_of_class(AlgebrasWithBasis(QQ).parent_class)
{'optional': ['one_basis', 'product_on_basis'], 'required': ['__contains__']}

Warning

The result above is not yet necessarily complete; many required methods in the categories are not yet marked as abstract_methods(). We also recommend browsing the documentation of this category: AlgebrasWithBasis.

Adding these methods, here is the minimal implementation of the group algebra:

sage: class MyCyclicGroupAlgebra(CombinatorialFreeModule):
....:
....:     def __init__(self, R, n, **keywords):
....:         self._group = Zmod(n)
....:         CombinatorialFreeModule.__init__(self, R, self._group,
....:             category=AlgebrasWithBasis(R), **keywords)
....:
....:     def product_on_basis(self, left, right):
....:         return self.monomial( left + right )
....:
....:     def one_basis(self):
....:         return self._group.zero()
....:
....:     def algebra_generators(self):
....:         return Family( [self.monomial( self._group(1) ) ] )
....:
....:     def _repr_(self):
....:         return "Jason's group algebra of %s over %s"%(self._group, self.base_ring())

Some notes about this implementation:

• Alternatively, we could have defined product instead of product_on_basis:

  ....:     # def product(self, left, right):
  ....:     #     return ## something ##
  

• For the sake of readability in this tutorial, we have stripped out all the documentation strings. Of course all of those should be present as in E.

• The purpose of **keywords is to pass down options like prefix to CombinatorialFreeModules.

Let us do some calculations:

sage: A = MyCyclicGroupAlgebra(QQ, 2, prefix='a') # or 4 or 5 or 11     ...
sage: a = A.basis();
sage: f = A.an_element();
sage: A, f
(Jason's group algebra of Ring of integers modulo 2 over Rational Field, 2*a[0] + 2*a[1])
sage: f * f
8*a[0] + 8*a[1]
sage: f.<tab>                                 # not tested
sage: f.is_idempotent()
False
sage: A.one()
a[0]
sage: x = A.algebra_generators().first() # Typically x,y,    ... = A.algebra_generators()
sage: [x^i for i in range(4)]
[a[0], a[1], a[0], a[1]]
sage: g = 2*a[1]; (f + g)*f == f*f + g*f
True

This seems to work fine, but we would like to put more stress on our implementation to shake potential bugs out of it. To this end, we will use TestSuite, a tool that performs many routine tests on our algebra for us.

Since we defined the class interactively, instead of in a Python module, those tests will complain about “pickling”. We can silence this error by making sage think that the class is defined in a module. We could also just ignore those failing tests for now or call TestSuite with the argument \(skip='_test_pickling')\):

sage: import __main__
sage: __main__.MyCyclicGroupAlgebra = MyCyclicGroupAlgebra

Ok, let’s run the tests:

sage: TestSuite(A).run(verbose=True)
running ._test_additive_associativity() . . . pass
running ._test_an_element() . . . pass
running ._test_associativity() . . . pass
running ._test_cardinality() . . . pass
running ._test_category() . . . pass
running ._test_characteristic() . . . pass
running ._test_construction() . . . pass
running ._test_distributivity() . . . pass
running ._test_elements() . . .
  Running the test suite of self.an_element()
  running ._test_category() . . . pass
  running ._test_eq() . . . pass
  running ._test_new() . . . pass
  running ._test_nonzero_equal() . . . pass
  running ._test_not_implemented_methods() . . . pass
  running ._test_pickling() . . . pass
  pass
running ._test_elements_eq_reflexive() . . . pass
running ._test_elements_eq_symmetric() . . . pass
running ._test_elements_eq_transitive() . . . pass
running ._test_elements_neq() . . . pass
running ._test_eq() . . . pass
running ._test_new() . . . pass
running ._test_not_implemented_methods() . . . pass
running ._test_one() . . . pass
running ._test_pickling() . . . pass
running ._test_prod() . . . pass
running ._test_some_elements() . . . pass
running ._test_zero() . . . pass

For more information on categories, see Elements, parents, and categories in Sage: a (draft of) primer:

sage: sage.categories.primer?                 # not tested

Review

We wanted to implement an algebra, so we:

1. Created the underlying vector space using CombinatorialFreeModule

2. Looked at sage.categories.<tab> to find an appropriate category

3. Loaded an example of that category, and used sage.misc.abstract_method.abstract_methods_of_class(), to see what methods we needed to write

4. Added the category information and other necessary methods to our class

5. Ran TestSuite to catch potential discrepancies

Exercises

1. Make a tiny modification to product_on_basis in “MyCyclicGroupAlgebra” to implement the dual of the group algebra of the cyclic group instead of its group algebra (so the product is now given by \(b_fb_g=\delta_{f,g}bf\)).

   Run the TestSuite tests (you may ignore the “pickling” errors). What do you notice?

   Fix the implementation of one and check that the TestSuite tests now pass.

   Add the Hopf algebra structure. Hint: look at the example:

   sage: C = HopfAlgebrasWithBasis(QQ).example()
   

2. Given a set \(S\), say:

   sage: S = Set([1,2,3,4,5])
   

   and a base ring, say:

   sage: R = QQ
   

   implement an \(R\)-algebra:

   sage: F = SubsetAlgebraOnFundamentalBasis(S, R)   # todo: not implemented
   

   with a basis (b_s)_{s\subset S} indexed by the subsets of S:

   sage: Subsets(S)
   Subsets of {1, 2, 3, 4, 5}
   

   and where the product is defined by \(b_s b_t = b_{s\cup t}\).

Morphisms

To better understand relationships between algebraic spaces, one wants to consider morphisms between them:

sage: A.module_morphism?                      # not tested
sage: A = MyCyclicGroupAlgebra(QQ, 2, prefix='a')
sage: B = MyCyclicGroupAlgebra(QQ, 6, prefix='b')
sage: A, B
(Jason's group algebra of Ring of integers modulo 2 over Rational Field, Jason's group algebra of Ring of integers modulo 6 over Rational Field)

sage: def func_on_basis(g):
....:     r"""
....:     This function is the 'brain' of a (linear) morphism
....:     from A --> B.
....:     The input is the index of basis element of the domain (A).
....:     The output is an element of the codomain (B).
....:     """
....:     if g==1: return B.monomial(Zmod(6)(3))# g==1 in the range A
....:     else:    return B.one()

We can now define a morphism that extends this function to \(A\) by linearity:

sage: phi = A.module_morphism(func_on_basis, codomain=B)
sage: f = A.an_element()
sage: f
2*a[0] + 2*a[1]
sage: phi(f)
2*b[0] + 2*b[3]

Exercise

Define a new free module In with basis indexed by the subsets of \(S\), and a morphism phi from In to F defined by

\[\phi(In_s) = \sum_{t\subset s} F_t\]

Diagonal and Triangular Morphisms

We now illustrate how to specify that a given morphism is diagonal or triangular with respect to some order on the basis, which means that the morphism is invertible and \(Sage\) is able to compute the inverse morphism automatically. Currently this feature requires the domain and codomain to have the same index set (in progress …).

sage: X = CombinatorialFreeModule(QQ, Partitions(), prefix='x'); x = X.basis();
sage: Y = CombinatorialFreeModule(QQ, Partitions(), prefix='y'); y = Y.basis();

A diagonal module morphism takes as argument a function whose input is the index of a basis element of the domain, and whose output is the coefficient of the corresponding basis element of the codomain:

sage: def diag_func(p):
....:     if len(p)==0: return 1
....:     else: return p[0]
....:
....:
sage: diag_func(Partition([3,2,1]))
3
sage: X_to_Y = X.module_morphism(diagonal=diag_func, codomain=Y)
sage: f = X.an_element();
sage: f
2*x[[]] + 2*x[[1]] + 3*x[[2]]
sage: X_to_Y(f)
2*y[[]] + 2*y[[1]] + 6*y[[2]]

Python fun fact: ~ is the inversion operator (but be careful with int’s!):

sage: ~2
1/2
sage: ~(int(2)) # in python this is the bitwise complement: ~x = -x-1
-3

Diagonal module morphisms are invertible:

sage: Y_to_X = ~X_to_Y
sage: f = y[Partition([3])] - 2*y[Partition([2,1])]
sage: f
-2*y[[2, 1]] + y[[3]]
sage: Y_to_X(f)
-x[[2, 1]] + 1/3*x[[3]]
sage: X_to_Y(Y_to_X(f))
-2*y[[2, 1]] + y[[3]]

For triangular morphisms, just like ordinary morphisms, we need a function that accepts as input the index of a basis element of the domain and returns an element of the codomain. We think of this function as representing the columns of the matrix of the linear transformation:

sage: def triang_on_basis(p):
....:     return Y.sum_of_monomials(mu for mu in Partitions(sum(p)) if mu >= p)
....:
sage: triang_on_basis([3,2])
y[[3, 2]] + y[[4, 1]] + y[[5]]
sage: X_to_Y = X.module_morphism(triang_on_basis, triangular='lower', unitriangular=True, codomain=Y)
sage: f = x[Partition([1,1,1])] + 2*x[Partition([3,2])];
sage: f
x[[1, 1, 1]] + 2*x[[3, 2]]

sage: X_to_Y(f)
y[[1, 1, 1]] + y[[2, 1]] + y[[3]] + 2*y[[3, 2]] + 2*y[[4, 1]] + 2*y[[5]]

Triangular module_morphisms are also invertible, even if X and Y are both infinite-dimensional:

sage: Y_to_X = ~X_to_Y
sage: f
x[[1, 1, 1]] + 2*x[[3, 2]]
sage: Y_to_X(X_to_Y(f))
x[[1, 1, 1]] + 2*x[[3, 2]]

For details, see ModulesWithBasis.ParentMethods.module_morphism() (and also sage.categories.modules_with_basis.TriangularModuleMorphism):

sage: A.module_morphism?                      # not tested

Exercise

Redefine the morphism phi from the previous exercise as a morphism that is triangular with respect to inclusion of subsets and define the inverse morphism. You may want to use the following comparison key as key argument to modules_morphism:

sage: def subset_key(s):
....:     """
....:     A comparison key on sets that gives a linear extension
....:     of the inclusion order.
....:
....:     INPUT:
....:
....:      - ``s`` -- set
....:
....:     EXAMPLES::
....:
....:         sage: sorted(Subsets([1,2,3]), key=subset_key)
....:         [{}, {1}, {2}, {3}, {1, 2}, {1, 3}, {2, 3}, {1, 2, 3}]
....:     """
....:     return (len(s), list(s))

Coercions

Once we have defined a morphism from \(X \to Y\), we can register it as a coercion. This will allow Sage to apply the morphism automatically whenever we combine elements of \(X\) and \(Y\) together. See http://sagemath.com/doc/reference/coercion.html for more information. As a training step, let us first define a morphism \(X\) to \(Y\), and register it as a coercion:

sage: def triang_on_basis(p):
....:     return Y.sum_of_monomials(mu for mu in Partitions(sum(p)) if mu >= p)

sage: triang_on_basis([3,2])
y[[3, 2]] + y[[4, 1]] + y[[5]]
sage: X_to_Y = X.module_morphism(triang_on_basis, triangular='lower', unitriangular=True, codomain=Y)
sage: X_to_Y.<tab>                            # not tested
sage: X_to_Y.register_as_coercion()

Now we can not only convert elements from \(X\) to \(Y\), but we can also do mixed arithmetic with these elements:

sage: Y(x[Partition([3,2])])
y[[3, 2]] + y[[4, 1]] + y[[5]]
sage: Y([2,2,1]) + x[Partition([2,2,1])]
2*y[[2, 2, 1]] + y[[3, 1, 1]] + y[[3, 2]] + y[[4, 1]] + y[[5]]

Exercise

Use the inverse of phi to implement the inverse coercion from F to In. Reimplement In as an algebra, with a product method making it use phi and its inverse.

A digression: new bases and quotients of symmetric functions

As an application, we show how to combine what we have learned to implement a new basis and a quotient of the algebra of symmetric functions:

sage: SF = SymmetricFunctions(QQ);  # A graded Hopf algebra
sage: h  = SF.homogeneous()         # A particular basis, indexed by partitions (with some additional magic)

So, \(h\) is a graded algebra whose basis is indexed by partitions. In more detail, h([i]) is the sum of all monomials of degree \(i\):

sage: h([2]).expand(4)
x0^2 + x0*x1 + x1^2 + x0*x2 + x1*x2 + x2^2 + x0*x3 + x1*x3 + x2*x3 + x3^2

and h(mu) = prod( h(p) for p in mu ):

sage: h([3,2,2,1]) == h([3]) * h([2]) * h([2]) * h([1])
True

Here we define a new basis \((X_\lambda)_\lambda\) by triangularity with respect to \(h\); namely, we set \(X_\lambda = \sum_{\mu\geq \lambda, |\mu|=|\nu|} h_\mu\):

sage: class MySFBasis(CombinatorialFreeModule):
....:     r"""
....:     Note: We would typically use SymmetricFunctionAlgebra_generic
....:     for this. This is as an example only.
....:     """
....:
....:     def __init__(self, R, *args, **kwargs):
....:         """ TODO: Informative doc-string and examples """
....:         CombinatorialFreeModule.__init__(self, R, Partitions(), category=AlgebrasWithBasis(R), *args, **kwargs)
....:         self._h = SymmetricFunctions(R).homogeneous()
....:         self._to_h = self.module_morphism( self._to_h_on_basis, triangular='lower', unitriangular=True, codomain=self._h)
....:         self._from_h = ~(self._to_h)
....:         self._to_h.register_as_coercion()
....:         self._from_h.register_as_coercion()
....:
....:     def _to_h_on_basis(self, la):
....:         return self._h.sum_of_monomials(mu for mu in Partitions(sum(la)) if mu >= la)
....:
....:     def product(self, left, right):
....:         return self( self._h(left) * self._h(right) )
....:
....:     def _repr_(self):
....:         return "Jason's basis for symmetric functions over %s"%self.base_ring()
....:
....:     @cached_method
....:     def one_basis(self):
....:         r""" Returns the index of the basis element that is equal to '1'."""
....:         return Partition([])
sage: X = MySFBasis(QQ, prefix='x'); x = X.basis(); h = SymmetricFunctions(QQ).homogeneous()
sage: f = X(h([2,1,1]) - 2*h([2,2]))  # Note the capital X
sage: f
x[[2, 1, 1]] - 3*x[[2, 2]] + 2*x[[3, 1]]
sage: h(f)
h[2, 1, 1] - 2*h[2, 2]
sage: f*f*f
x[[2, 2, 2, 1, 1, 1, 1, 1, 1]] - 7*x[[2, 2, 2, 2, 1, 1, 1, 1]] + 18*x[[2, 2, 2, 2, 2, 1, 1]]
- 20*x[[2, 2, 2, 2, 2, 2]] + 8*x[[3, 1, 1, 1, 1, 1, 1, 1, 1, 1]]
sage: h(f*f)
h[2, 2, 1, 1, 1, 1] - 4*h[2, 2, 2, 1, 1] + 4*h[2, 2, 2, 2]

We now implement a quotient of the algebra of symmetric functions obtained by killing any monomial symmetric function \(m_\lambda\) such that the first part of \(\lambda\) is greater than \(k\). See Sets.SubcategoryMethods.Subquotients() for more details about implementing quotients:

sage: class MySFQuotient(CombinatorialFreeModule):
....:     r"""
....:     The quotient of the ring of symmetric functions by the ideal generated
....:     by those monomial symmetric functions whose part is larger than some fixed
....:     number ``k``.
....:     """
....:     def __init__(self, R, k, prefix=None, *args, **kwargs):
....:         CombinatorialFreeModule.__init__(self, R,
....:             Partitions(NonNegativeIntegers(), max_part=k),
....:             prefix = 'mm',
....:             category = Algebras(R).Graded().WithBasis().Quotients(), *args, **kwargs)
....:
....:         self._k = k
....:         self._m = SymmetricFunctions(R).monomial()
....:
....:         self.lift = self.module_morphism(self._m.monomial)
....:         self.retract = self._m.module_morphism(self._retract_on_basis, codomain=self)
....:
....:         self.lift.register_as_coercion()
....:         self.retract.register_as_coercion()
....:
....:     def ambient(self):
....:         return self._m
....:
....:     def _retract_on_basis(self, mu):
....:         r"""
....:         Takes the index of a basis element of a monomial
....:         symmetric function, and returns the projection of that
....:         element to the quotient.
....:         """
....:         if len(mu) > 0 and mu[0] > self._k:
....:             return self.zero()
....:         return self.monomial(mu)
....:
sage: MM = MySFQuotient(QQ, 3)
sage: mm = MM.basis()
sage: m = SymmetricFunctions(QQ).monomial()
sage: P = Partition
sage: g = m[P([3,2,1])] + 2*m[P([3,3])] + m[P([4,2])]; g
m[3, 2, 1] + 2*m[3, 3] + m[4, 2]
sage: f = MM(g); f
mm[[3, 2, 1]] + 2*mm[[3, 3]]
sage: m(f)
m[3, 2, 1] + 2*m[3, 3]

sage: (m(f))^2
8*m[3, 3, 2, 2, 1, 1] + 12*m[3, 3, 2, 2, 2] + 24*m[3, 3, 3, 2, 1] + 48*m[3, 3, 3, 3]
+ 4*m[4, 3, 2, 2, 1] + 4*m[4, 3, 3, 1, 1] + 14*m[4, 3, 3, 2] + 4*m[4, 4, 2, 2]
+ 4*m[4, 4, 3, 1] + 6*m[4, 4, 4] + 4*m[5, 3, 2, 1, 1] + 4*m[5, 3, 2, 2]
+ 12*m[5, 3, 3, 1] + 2*m[5, 4, 2, 1] + 6*m[5, 4, 3] + 4*m[5, 5, 1, 1] + 2*m[5, 5, 2]
+ 4*m[6, 2, 2, 1, 1] + 6*m[6, 2, 2, 2] + 6*m[6, 3, 2, 1] + 10*m[6, 3, 3] + 2*m[6, 4, 1, 1] + 5*m[6, 4, 2] + 4*m[6, 5, 1] + 4*m[6, 6]

sage: f^2
8*mm[[3, 3, 2, 2, 1, 1]] + 12*mm[[3, 3, 2, 2, 2]] + 24*mm[[3, 3, 3, 2, 1]] + 48*mm[[3, 3, 3, 3]]

sage: (m(f))^2 - m(f^2)
4*m[4, 3, 2, 2, 1] + 4*m[4, 3, 3, 1, 1] + 14*m[4, 3, 3, 2] + 4*m[4, 4, 2, 2] + 4*m[4, 4, 3, 1] + 6*m[4, 4, 4] + 4*m[5, 3, 2, 1, 1] + 4*m[5, 3, 2, 2] + 12*m[5, 3, 3, 1] + 2*m[5, 4, 2, 1] + 6*m[5, 4, 3] + 4*m[5, 5, 1, 1] + 2*m[5, 5, 2] + 4*m[6, 2, 2, 1, 1] + 6*m[6, 2, 2, 2] + 6*m[6, 3, 2, 1] + 10*m[6, 3, 3] + 2*m[6, 4, 1, 1] + 5*m[6, 4, 2] + 4*m[6, 5, 1] + 4*m[6, 6]

sage: MM( (m(f))^2 - m(f^2) )
0

Implementing algebraic structures with several realizations

We now return to the subset algebra and use it as an example to show how to implement several different bases for an algebra with automatic coercions between the different bases. We have already implemented three bases for this algebra: the F, In, and Out bases, as well as coercions between them. In real calculations it is convenient to tie these parents together by implementing an object A that models the abstract algebra itself. Then, the parents F, In and Out will be realizations of A, while A will be a parent with realizations. See Sets().WithRealizations for more information about the expected user interface and the rationale.

Here is a brief template highlighting the overall structure:

class MyAlgebra(Parent, UniqueRepresentation):
    def __init__(self, R, ...):
        category = Algebras(R).Commutative()
        Parent.__init__(self, category=category.WithRealizations())
        # attribute initialization, construction of the morphisms
        # between the bases, ...

    class Bases(Category_realization_of_parent):
        def super_categories(self):
            A = self.base()
            category = Algebras(A.base_ring()).Commutative()
            return [A.Realizations(), category.Realizations().WithBasis()]

        class ParentMethods:
            r"""Code that is common to all bases of the algebra"""

        class ElementMethods:
            r"""Code that is common to elements of all bases of the algebra"""

    class FirstBasis(CombinatorialFreeModule, BindableClass):
        def __init__(self, A):
            CombinatorialFreeModule.__init__(self, ..., category=A.Bases())

        # implementation of the multiplication, the unit, ...

    class SecondBasis(CombinatorialFreeModule, BindableClass):
        def __init__(self, A):
            CombinatorialFreeModule.__init__(self, ..., category=A.Bases())

        # implementation of the multiplication, the unit, ...

The class MyAlgebra implements a commutative algebra A with several realizations, which we specify in the constructor of MyAlgebra. The two bases classes MyAlgebra.FirstBasis and MyAlgebra.SecondBasis implement different realizations of A that correspond to distinguished bases on which elements are expanded. They are initialized in the category MyAlgebra.Bases of all bases of A, whose role is to factor out their common features. In particular, this construction says that they are:

• realizations of A

• realizations of a commutative algebra, with a distinguished basis

Note

There is a bit of redundancy here: given that A knows it is a commutative algebra with realizations the infrastructure could, in principle, determine that its realizations are commutative algebras. If this was done then it would be possible to implement \(Bases.super_categories\) by returning:

[A.Realizations().WithBasis()]

However, this has not been implemented yet.

Note

Inheriting from BindableCass just provides syntactic sugar: it makes MyAlgebras().FirstBasis() a shorthand for MyAlgebras.FirstBasis(MyAlgebras().FirstBasis()) (binding behavior). The class Bases inherits this binding behavior from Category_realization_of_parent , which is why we can write MyAlgebras().Bases instead of MyAlgebras.Bases(MyAlgebras())

Note

More often than not, the constructors for all of the bases will be very similar, if not identical; so we would want to factor it out. Annoyingly, the natural approach of putting the constructor in Bases.ParentMethods does not work because this is an abstract class whereas the constructor handles the concrete implementation of the data structure. Similarly, it would be better if it was only necessary to specify the classes the bases inherit from once, but this can’t code go into Bases for the same reason.

The current recommended solution is to have an additional class Basis that factors out the common concrete features of the different bases:

...

class Basis(CombinatorialFreeModule, BindableClass):
    def __init__(self, A):
        CombinatorialFreeModule.__init__(self, ..., category=A.Bases())

class FirstBasis(Basis):
    ...

class SecondBasis(Basis):
    ...

This solution works but it is not optimal because to share features between the two bases code needs to go into two locations, Basis and Bases, depending on whether they are concrete or abstract, respectively.

We now urge the reader to browse the full code of the following example, which is meant as a complete template for constructing new parents with realizations:

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

sage: A??                                     # not implemented

Review

Congratulations on reading this far!

We have now been through a complete tour of the features needed to implement an algebra with several realizations. The infrastructure for realizations is not tied specifically to algebras; what we have learned applies mutatis mutandis in full generality, for example for implementing groups with several realizations.