I will summarise what I have learnt from Nicolas Thiery:
If M is a monoid then the monoid algebra is constructing using
Monoid().Algebra() by, say, M.algebra(QQ).
For groups the construction GroupAlgebra is being replaced by a similar
G.algebra().
This is part of an ongoing program to replace functions using FormalSums by
functions using CombinatorialFreeModule()
The directed graph C.category_graph() can be drawn using graphviz.
Specifically
sage: C = Monoids().Algebras(QQ)
sage: G = C.category_graph()
sage: G.set_latex_options(format="dot2tex")
sage: view(G, viewer="pdf", tightpage=True)
I have not personally tested this as I would have to install the optional
packages graphviz and dot2tex
A group algebra is constructed as a Hopf algebra but a monoid algebra is not
constructed as a bialgebra
although it would be straightforward for it to do so.
I am still confused about categories and parents and am unclear on how to
implement a new parent.
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