Lattices of dominions in quasivarieties of Abelian groups

Dominions of universal algebras and projective properties

“…Proof. For an Abelian quasivariety M, the statement of the theorem follows immediately from [10]. Note that if A = E is a trivial group then it follows by the definition of a dominion that dom M B (E) = E for every group B ∈ M. We assume that M is non-Abelian.…”

“…The advisability of studying dominions in quasivarieties of universal algebras was grounded in [7] by the observation that according to [9], among axiomatizable classes, only quasivarieties possess a complete theory of defining relations, which allows a free product with amalgamated subalgebra to be defined. Dominions were discussed at length for quasivarieties of Abelian groups in [10,11]; lattices of dominions, in [7,12].…”

Dominions in Solvable Groups

“…Note that dominions were thoroughly investigated for quasivarieties of Abelian groups [3][4][5][6]. Dominions in the class of nilpotent groups were dealt with in a series of papers [7][8][9].…”

Lattices of dominions of universal algebras

“…The concept of a dominion was introduced in [ Furthermore, the concept of a dominion was dealt with in various classes of algebras [2][3][4] (see also the bibliography in [5]). In particular, it was established that dominions are closely related to amalgams (for details, see [2]).…”

“…4.4], it was found out that there exists a relationship between lattices of quasivarieties and lattices of dominions. In [4], the latter lattices were thoroughly studied for the case of Abelian groups.…”