Abstract
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.
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Presented by G. McNulty.
Dedicated to Brian Davey on the occasion of his 65th birthday
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Quackenbush, R.W. Skolemizing algebraically closed universal classes of algebras. Algebra Univers. 74, 361–380 (2015). https://doi.org/10.1007/s00012-015-0350-3
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DOI: https://doi.org/10.1007/s00012-015-0350-3