Incidence Algebras

sage.combinat.posets.incidence_algebras.IncidenceAlgebra

The incidence algebra of a poset.

Let \(P\) be a poset and \(R\) be a commutative unital associative ring. The incidence algebra \(I_P\) is the algebra of functions \(\alpha \colon P \times P \to R\) such that \(\alpha(x, y) = 0\) if \(x \not\leq y\) where multiplication is given by convolution:

\[(\alpha \ast \beta)(x, y) = \sum_{x \leq k \leq y} \alpha(x, k) \beta(k, y).\]

This has a natural basis given by indicator functions for the interval \([a, b]\), i.e. \(X_{a,b}(x,y) = \delta_{ax} \delta_{by}\). The incidence algebra is a unital algebra with the identity given by the Kronecker delta \(\delta(x, y) = \delta_{xy}\). The Möbius function of \(P\) is another element of \(I_p\) whose inverse is the \(\zeta\) function of the poset (so \(\zeta(x, y) = 1\) for every interval \([x, y]\)).

Todo

Implement the incidence coalgebra.

REFERENCES:

sage.combinat.posets.incidence_algebras.ReducedIncidenceAlgebra

The reduced incidence algebra of a poset.

The reduced incidence algebra \(R_P\) is a subalgebra of the incidence algebra \(I_P\) where \(\alpha(x, y) = \alpha(x', y')\) when \([x, y]\) is isomorphic to \([x', y']\) as posets. Thus the delta, Möbius, and zeta functions are all elements of \(R_P\).