Gradings on incidence algebras and their graded polynomial identities (Q2661727)

Let \(G\) be a group, \(P\) a partially ordered set which is locally finite, and let \(I(P,F)\) be the incidence algebra of \(P\) over a field \(F\). The authors study elementary \(G\)-gradings on \(I(P,F)\). In case \(P\) is a finite poset, and \(G\) a finite group they produce a formula that counts how many such gradings on \(I(P,F)\) there are. Furthermore, assume \(P\) is a bounded poset, and \(F\) a field of characteristic 0. Let two \(G\)-gradings on \(I(P,F)\) satisfy the same graded polynomial identities. The authors prove that if these two gradings satisfy the same graded identities and if the automorphism group of \(P\) acts transitively on the maximal chains of \(P\), then the two gradings determine isomorphic graded algebras. Moreover the transitivity condition cannot be dropped as the authors show. Reviewer's remark: In a recent paper, [\textit{E. A. Santulo jun.} et al., J. Algebra 544, 302--328 (2020; Zbl 1436.16051)] good and elementary gradings on incidence algebras were studied assuming \(P\) finite. In the latter paper, under the assumption of \(G\) being abelian, a very tight description of the gradings was obtained under some mild restrictions on the field \(F\).