Abstract
Let M be any quasivariety of Abelian groups, \(dom_G^M\)(H) be the dominion of a subgroup H of a group G in M, and Lq(M) be the lattice of subquasivarieties of M. It is proved that \(dom_G^M\)(H ) coincides with a least normal subgroup of the group G containing H, the factor group with respect to which is in M. Conditions are specified subject to which the set L(G,H,M) = {\(dom_G^N\)(H) | N ∈ Lq(M)} forms a lattice under set-theoretic inclusion and the map ϕ : Lq(M) → L(G,H,M) such that ϕ(N) = \(dom_G^N\)(H) for any quasivariety N ϕ Lq(M)is an antihomomorphism of the lattice L q (M) onto the lattice L(G, H, M).
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Translated from Algebra i Logika, Vol. 44, No. 2, pp. 238–251, March–April, 2005.
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Shakhova, S.A. Lattices of dominions in quasivarieties of Abelian groups. Algebr Logic 44, 132–139 (2005). https://doi.org/10.1007/s10469-005-0014-z
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DOI: https://doi.org/10.1007/s10469-005-0014-z