For a semigroup S, the set of all subsemigroups of S × S, with the operations of composition and involution and the relation of set-theoretic inclusion, forms the bundle of correspondences of S, denoted by b S. A semigroup S is called b -determined if for any semigroup T, b S ≅ T if and only if S ≅ T or S ≅ T opp. The b -determinability of semilattices was established by G. I. Z̆itomirskiĭ ( Mat. Sb. 82, No. 2 (1970) , pp. 163–174) and that of abelian groups, nonperiodic groups, and nonperiodic commutative Clifford semigroups by D. A. Bredihin ( in “Studies in Algebra,” No. 4, pp. 3–12, Saratov Univ. Press, Saratov, USSR, 1974 ). Here the class of those inverse semigroups which are known to be determined by their bundles of correspondences is substantially enlarged. Our major goal is to prove that any fundamental inverse semigroup is b -determined. Then we study b -isomorphisms of nonperiodic semilattices of cancellative monoids (in particular, of nonperiodic Clifford semigroups) and describe new classes of b -determined semigroups of that type.