Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class {ie304-01}) is the set of all elements a ∈ A such that every pair of homomorphisms f, g: A → ∈ {ie304-02} satisfies the following: if f and g coincide on H, then f(a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group 〈H, a〉 generated by one element modulo H.
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Translated from Algebra i Logika, Vol. 47, No. 5, pp. 541–557, September–October, 2008.
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Budkin, A.I. Dominions of universal algebras and projective properties. Algebra Logic 47, 304–313 (2008). https://doi.org/10.1007/s10469-008-9029-6
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DOI: https://doi.org/10.1007/s10469-008-9029-6