Lattices of dominions of universal algebras

• Autores: A. I. Budkin
• Localización: Algebra and logic, ISSN 0002-5232, Vol. 46, Nº. 1, 2007, págs. 16-27
• Idioma: inglés
• DOI: 10.1007/s10469-007-0002-6
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• Resumen

  • 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.