# Obtaining the reduced incidence algebra in QPA

Given a finite poset $$P$$ (we can assume it is connected), the reduced incidence algebra of $$P$$ is the subalgebra of the incidence algebra of $$P$$ consisting of functions constant on isomorphic intervals.

The incidence algebra is isomorphic to the quiver algebra with quiver the Hasse poset of $$P$$ and the relations are such that paths of the same start and end get identified.

Question: Is it possible with the GAP-package QPA to obtain the reduced incidence algebra (as quiver and relations) of a poset in an easy way from the quiver of the incidence algebra?

Easy way could mean that one uses only functions that are already available in QPA, for example taking endomorphism rings of module or quotient algebras (and combinations). I can not think of a method how to do it but maybe I overlook something simple.

• Computing the quiver should be easy. It is a basic local algebra so you just need the dimension of the radical modulo the radical squared. The radical is the set of strictly upper triangular elements. – Benjamin Steinberg Jan 5 at 14:15
• There are a few questions on QPA on the MSE site, some answered by one of its authors - perhaps asking there would be more appropriate. For package-specific questions, getting in touch with authors is usually most efficient. – Alexander Konovalov Jan 5 at 20:19
• @AlexanderKonovalov Thanks. The question here could be also purely mathematical in case there is a nice way to write the reduced incidence algebra using existing/standard operations. So there is a good chance someone might answer it even without knowledge of QPA. – Mare Jan 5 at 20:25
• Sagemath has a lot of poset algorithms implemented. sagemath.org – Dima Pasechnik Jan 6 at 5:50
• doc.sagemath.org/html/en/reference/algebras/sage/combinat/… mentions reduced algebra computing. – Dima Pasechnik Jan 6 at 5:53