We obtain two new characterizations of separativity of re- finement monoids, in terms of comparability-type conditions. As appli- cations, we get several equivalent conditions of separativity for exchange ideals. A commutative monoid (M,+,0) has a refinement (or is a refinement mono- id) if, for all a,b,c and d in M, the equation a+b = c+d implies the existence of a1,b1,c1,d1 2 M such that a = a1+d1,b = b1+c1,c = a1+b1 and d = d1+c1. These equations are represented in the form of a refinement matrix: a b c d i a1 d1 b 1 c1 ¢ . Refinement monoids have been extensively studied in recent years (cf. (4) and (7)). A commutative monoid M is separative if, for all a,b 2 M, 2a = a+b = 2b implies a = b. Separativity is a weak form of cancellativity for commutative monoids. Many authors have studied separative refinement monoids from var- ious view-points (see (3-4) and (6-7)). In this article, we get two new charac- terizations of separative refinement monoids. We prove that every separative refinement monoid can be characterized by a certain sort of comparability. Also we introduce the concept of refinement extensions of a refinement monoid. We see that every separative refinement monoid can be characterized by such re- finement extensions. Let I be an ideal of a ring R. We use FP(I) to denote the class of finitely generated projective right R-modules P with P = PI and V (I) to denote the monoid of isomorphism classes of objects from FP(I). Fol- lowing Ara et al. (see (3)), an exchange ideal I of a ring R is separative if V (I) is a separative refinement monoid, that is, for any A,B,C 2 FP(I), A ' A » A ' B » B ' B =) A » B. We say that R is a separative ring if R is separative as an ideal of R. Separativity plays a key role in the direct sum decomposition theory of ex- change rings. It seems rather likely that non-separative exchange rings should exist. We say that an exchange ring R satisfies the comparability axiom pro- vided that, for any finitely generated projective right R-modules A and B, either A. ' B or B. ' A. In (7, Theorem 3.9), Pardo showed that every