  Given a finite set E and an operator sigma:2^{E}-->2^{E}, two subsets X,Y of
the ground set E are cospanning if sigma(X)=sigma(Y) (Korte, Lovasz, Schrader;
1991). We investigate cospanning relations on violator spaces. A notion of a
violator space was introduced in (Gartner, Matousek, Rust, Skovrovn; 2008) as a
combinatorial framework that encompasses linear programming and other geometric
optimization problems. Violator spaces are defined by violator operators. We
introduce co-violator spaces based on contracting operators known also as
choice functions. Let alpha,beta:2^{E}-->2^{E} be a violator operator and a
co-violator operator, respectively. Cospanning characterizations of violator
spaces allow us to obtain some new properties of violator operators,
co-violator operators, and their interconnections. In particular, we show that
uniquely generated violator spaces enjoy so-called Krein-Milman properties,
i.e., alpha(beta(X))=alpha(X) and beta(alpha}(X))=beta(X) for every subset X of
the ground set E.
