We prove that a variety V is a discriminator variety if and only if V has the Fraser-Horn property and every member of V is representable as a Boolean product whose factors are directly indecomposable or trivial. A variety has the Fraser-Horn property [3] if every congruence on a product Al x A2 is of the form 01 x 02. Examples of varieties with this property are congruence distributive varieties and congruence modular varieties in which no non-trivial member has a trivial subalgebra [8]. In this paper we prove the following: Theorem. For a variety V the following are equivalent: (a) V is a discriminator variety. (b) V has the Fraser-Horn property and every member of V is representable as a Boolean product whose factors are directly indecomposable or trivial. We remit to [2] for notation and basic facts on Boolean products and discriminator varieties. The concept of decomposition operation plays a deep role in the proof. A decomposition operation on an algebra A is a homomorphism d: A x A -* A satisfying: d(x, x) x, d(d(x, y), z) d(x, z) d(x, d(y, z)). Given a pair 0, o of complementary factor congruences we have associated a decomposition operation defined by do,6(a, b) = the unique c E A such that (c, a) E 0 and (c, b) E 6. Reciprocally, given a decomposition operation d, the relations Od = {(x)y): d(x,y) = y} and ad = {(x,y) : d(x,y) = x} are a pair of complementary factor congruences. These maps are mutually inverse [4]. For an algebra A let A denote the diagonal congruence on A. Proof of the Theorem. (a) =>(b) Since V is congruence distributive it has the FraserHorn property [1]. The Boolean representation is the well known result of BulmanFleming, Keimel and Werner [2]. (b)z=>(a) Since V has the Fraser-Horn property, we have that: (1) For every A E V, the set FC(A) of all factor congruences forms a Boolean sublattice of the congruence lattice of A. (2) If Ao : A -* B is an onto homomorphism and d is a decomposition operation on A, then the equation (o(d) ((o(a), (o(b)) = (o(d(a, b)) defines a decomposition operation (o(d) on B. Received by the editors March 2, 1999 and, in revised form, May 14, 1999. 2000 Mathematics Subject Classification. Primary 08A05, 08A30, 08A40, 08B10, 06E15. The author's research was supported by CONICOR and SECYT (UNC). (?)2000 American Mathematical Society