Idempotence preserving maps on spaces of triangular matrices

Suppose F is an arbitrary field. Let |F| be the number of the elements of F. Let Tn(F) be the space of all n×n upper-triangular matrices over F. A map Ψ : Tn(F) → Tn(F) is said to preserve idempotence if A - λB is idempotent if and only if Ψ(A) - λΨ(B) is idempotent for any A, B ∈ Tn(F) and λ ∈ F. It is shown that: when the characteristic of F is not 2, |F| > 3 and n ≥ 3, Ψ : Tn(F) → Tn(F) is a map preserving idempotence if and only if there exists an invertible matrix P ∈ Tn(F) such that either Φ(A) = PAP-1 for every A ∈ Tn(F) or Ψ (A) = PJAtJP-1 for every A ∈ Tn (F), where J = ∑i=1n E i,n+1-iand Eij is the matrix with 1 in the (i,j)th entry and 0 elsewhere. © 2007 Korean Society for Computational & Applied Mathematics.