Geodesic
Lattices of Dominions in Quasivarieties of Abelian~Groups
Algebra i logika, Tome 44 (2005) no. 2, pp. 238-251.

Voir la notice de l'article provenant de la source Math-Net.Ru

Let $\mathcal M$ be any quasivariety of Abelian groups, $\operatorname{dom}^{\mathcal M}_G(H)$ be the dominion of a subgroup $H$ of a group $G$ in $\mathcal M$, and $L_q(\mathcal M)$ be the lattice of subquasivarieties of $\mathcal M$. It is proved that $\operatorname{dom}^{\mathcal M}_G(H)$ coincides with a least normal subgroup of the group $G$ containing $H$, the factor group with respect to which is in $\mathcal M$. Conditions are specified subject to which the set $L(G,H,\mathcal M)=\{\operatorname{dom}^{\mathcal N}_G(H)\mid\mathcal N\in L_q(\mathcal M)\}$ forms a lattice under set-theoretic inclusion and the map $\varphi\colon L_q(\mathcal M)\rightarrow L(G,H,\mathcal M)$ such that $\varphi (\mathcal N)=\operatorname{dom}^{\mathcal N}_G(H)$ for any quasivariety $\mathcal N\in L_q(\mathcal M)$ is an antihomomorphism of the lattice $L_q(\mathcal M)$ onto the lattice $L(G,H,\mathcal M)$.
Keywords: quasivariety, dominion, lattice
Mots-clés : group. 
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     title = {Lattices of {Dominions} in {Quasivarieties} of {Abelian~Groups}},
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S. A. Shakhova. Lattices of Dominions in Quasivarieties of Abelian~Groups. Algebra i logika, Tome 44 (2005) no. 2, pp. 238-251. http://geodesic.mathdoc.fr/item/AL_2005_44_2_a5/

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