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We prove that every distributive algebraic lattice with at most ℵ1 compact elements is isomorphic to the normal subgroup lattice of some group and to the submodule lattice of some right module. The ℵ1 bound is optimal, as we find a distributive algebraic lattice D with ℵ2 compact elements that is not isomorphic to the congruence lattice of any algebra with almost permutable congruences (hence neither...
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