Linear maps preserving products equal to primitive idempotents of an incidence algebra

Abstract

Let A, B be algebras and a∈A, b∈B a fixed pair of elements. We say that a map φ:A→Bpreserves products equal to a and b if for all a1,a2∈A the equality a1a2=a implies φ(a1)φ(a2)=b. In this paper we study bijective linear maps φ:I(X,F)→I(X,F) preserving products equal to primitive idempotents of I(X,F), where I(X,F) is the incidence algebra of a finite connected poset X over a field F. We fully characterize the situation, when such a map φ exists, and whenever it does, φ is either an automorphism of I(X,F) or the negative of an automorphism of I(X,F).