The incidence algebra of a group reduced partially ordered set

Abstract

Let (P,⩽) be a locally finite partially ordered set with no infinite descending chains, let G be a finite subgroup of the automorphism group of P and let P(G) be the set of orbits of P under the action of G. Define a partial ordering ⩽_G on P(G) by saying that an orbit A is less than or equal to an orbit B if there exist elements a ε A and b ε B with a⩽b. We show that I (P(G)), the incidence algebra of P(G), is the homomorphic image of a subalgebra of I(P) via a Polya-type homomorphism. We compute a natural inverse image for each function of I(P(G)) and in particular for the Möbius function of P(G).