Abstract: All subsets P of an irreducible affine root system R such that both P and R\P are closed under addition of roots are classified. It is shown that if f mapping R to R' is a bijection of root systems such that f and its inverse preserve closed sets and the irreducible components of R and R' are finite or affine with at most one component of type A1, then f is an isomorphism of root systems.
This article appeared in Journal of Pure and Applied Algebra 131 (1998), 133-142.