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Algebraically closed abelian groups are precisely the divisible abelian groups; the underlying groups of rational vector spaces are precisely the divisible abelian groups. That is, rational vector spaces are the skolemization of algebraically closed abelian groups, obtained by adding scalar multiplication by each rational to the set of operations. Algebraically closed bounded distributive lattices are precisely the complemented distributive lattices; the underlying lattices of boolean algebras are precisely the complemented distributive lattices. That is, boolean algebras are the skolemization of algebraically closed bounded distributive lattices, obtained by adding complementation to the set of operations. We explore this idea for arbitrary universal classes of algebras and focus particularly on the case of meet semilattices. The algebraically closed meet semilattices are precisely the distributive meet semilattices; their skolemization seems not to have been discussed previously in the literature. We discuss it here but reach no firm solution. [ABSTRACT FROM AUTHOR]

Abstract: According to Markov (1946) , a subset of an abelian group G of the form , for some integer n and some element , is an elementary algebraic set; finite unions of elementary algebraic sets are called algebraic sets. We prove that a subset of an abelian group G is algebraic if and only if it is closed in every precompact (= totally bounded) Hausdorff group topology on G. The family of all algebraic sets of an abelian group G forms the family of closed subsets of a unique Noetherian topology on G called the Zariski, or verbal, topology of G; see Bryant (1977) . We investigate the properties of this topology. In particular, we show that the Zariski topology is always hereditarily separable and Fréchet–Urysohn. For a countable family of subsets of an abelian group G of cardinality at most the continuum, we construct a precompact metric group topology on G such that the -closure of each member of coincides with its -closure. As an application, we provide a characterization of the subsets of G that are -dense in some Hausdorff group topology on G, and we show that such a topology, if it exists, can always be chosen so that it is precompact and metric. This provides a partial answer to a long-standing problem of Markov (1946) . [ABSTRACT FROM AUTHOR]