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 → B preserves products equal to a and b if for all a 1 , a 2 ∈ A the equality a 1 a 2 = a implies ϕ(a 1 )ϕ(a 2 ) = 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 ).