Lattices of dominions of universal algebras |
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Authors: | A I Budkin |
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Institution: | (1) Pavlovskii road, 60a-168, Barnaul, 656064, Russia |
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Abstract: | We fix a universal algebra A and its subalgebra H. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that any pair of homomorphisms f, g: A → M ∈ M satisfies the following: if f and g coincide on H then f(a) = g(a). In association with every quasivariety, therefore, is
a dominion of H in A. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions
is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety
in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that
quasivariety.
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Translated from Algebra i Logika, Vol. 46, No. 1, pp. 26–45, January–February, 2007. |
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Keywords: | dominion lattice of dominions quasivariety |
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