Anti-elementarity is a strong way of ensuring that a class of structures , in a given first-order language, is not closed under elementary equivalence with respect to any infinitary language of the form $\mathscr{L}_{\infty\lambda}$. We prove that many naturally defined classes are anti-elementary, including the following: $\bullet$ the class of all lattices of finitely generated convex $\ell$-subgroups of members of any class of $\ell$-groups containing all Archimedean $\ell$-groups; $\bullet$ the class of all semilattices of finitely generated $\ell$-ideals of members of any nontrivial quasivariety of $\ell$-groups; $\bullet$ the class of all Stone duals of spectra of MV-algebras-this yields a negative solution for the MV-spectrum Problem; $\bullet$ the class of all semilattices of finitely generated two-sided ideals of rings; $\bullet$ the class of all semilattices of finitely generated submodules of modules; $\bullet$ the class of all monoids encoding the nonstable K$_0$-theory of 查看全文>>