Hopfian $\mathrm{\ell }$-groups, MV-algebras and AF C${}^{\ast }$-algebras

An algebra is said to be hopfian if it is not isomorphic to a proper quotient of itself. We describe several classes of hopfian and of non-hopfian unital lattice-ordered abelian groups and MV-algebras. Using Elliott classification and $K_0$-theory, we apply our results to other related structures, notably the Farey-Stern-Brocot AF C${}^{\ast }$-algebra and all its primitive quotients, including the Behnke-Leptin C${}^{\ast }$-algebras ${\mathcal{A}}_{k,q}$.