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An algebra is said to be hopfian if it is not isomorphic to a proper quotientof itself. We describe several classes of hopfian and of non-hopfian unitallattice-ordered abelian groups and MV-algebras. Using Elliott classificationand $K_0$-theory, we apply our results to other related structures, notably theFarey-Stern-Brocot AF C$^*$-algebra and all its primitive quotients, includingthe Behnke-Leptin C$^*$-algebras $\mathcal A_{k,q}$.

Hopfian $\ell$-groups, MV-algebras and AF C$^*$-algebras