Geodesic
Distributivity Conditions for Lattices of Dominions in Quasivarieties of Abelian Groups
Algebra i logika, Tome 45 (2006) no. 4, pp. 484-499.

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Let $\mathcal{M}$ be any quasivariety of Abelian groups, $L_{q}(\mathcal{M})$ be a subquasivariety lattice of $\mathcal{M}$, ${\rm dom}^{\mathcal{M}}_{G}(H)$ be the dominion of a subgroup $H$ of a group $G$ in $\mathcal{M}$, and $G/{\rm dom}^{\mathcal{M}}_{G}(H)$ be a finitely generated group. It is known that the set $L(G,H,\mathcal{M})=\{{\rm dom}^{\mathcal{N}}_{G}(H)\mid \mathcal{N}\in L_{q}(\mathcal{M})\}$ forms a lattice w.r.t. set-theoretic inclusion. We look at the structure of ${\rm dom}^{\mathcal{M}}_{G}(H)$. It is proved that the lattice $L(G,H,\mathcal{M})$ is semidistributive and necessary and sufficient conditions are specified for its being distributive.
Mots-clés : group
Keywords: dominion, quasivariety, lattice. 
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     author = {S. A. Shakhova},
     title = {Distributivity {Conditions} for {Lattices} of {Dominions} in {Quasivarieties} of {Abelian} {Groups}},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2006_45_4_a6/}
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S. A. Shakhova. Distributivity Conditions for Lattices of Dominions in Quasivarieties of Abelian Groups. Algebra i logika, Tome 45 (2006) no. 4, pp. 484-499. http://geodesic.mathdoc.fr/item/AL_2006_45_4_a6/

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