Since closed cones C in finite dimensional real vector spaces V play the role of “Lie algebras” for finite dimensional compact abelian semigroups, we discuss the structure of congruence relations on such cones. We show in particular that a monotone closed congruence on such a cone C is entirely determined by a countable collection of closed ideals of C and a countable collection of linear subspaces of V (see theorem 3.3).