Asymptotic ordering of distribution functions and convolution semigroups

It is well-known that the set H of distribution functions on [0,∞] supplied with the convolution product * is a semigroup. We introduce a preorder on (H,*) with attractive algebraic properties. The corresponding equivalence relation ≈^W extends the concept of tail-equivalence of distribution functions. We show that the idempotents of the factorsemigroup H _W form a subsemigroup of H _W.