Dominions in quasivarieties of universal algebras

Abstract

The dominion of a subalgebra H in an universal algebra A (in a class M) is the set of all elements a ∈ A such that for all homomorphisms f, g :We investigate the connection between dominions and quasivarieties. We show that if a class M is closed under ultraproducts, then the dominion in M is equal to the dominion in a quasivariety generated by M. Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.

Dominions in Solvable Groups

“…Interest in research on dominions is primarily conditioned by their being densely linked with free constructions in quasivarieties of universal algebras and with amalgams (see [1] and bibliography in [2]). …”

Dominions of universal algebras and projective properties

“…According to [1] [2][3][4][5][6] (see also bibliography in [7]). In particular, it was established that there exists a close relationship between dominions and amalgams (for details, see [8]).…”

“…In particular, it was established that there exists a close relationship between dominions and amalgams (for details, see [8]). 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].…”

Lattices of dominions in quasivarieties of Abelian groups

“…Among axiomatizable classes, however, only quasivarieties were found to enjoy a complete theory of defining relations, which allows of determining a free amalgamated product in these, given any amalgam [6; see also 7]. This was an important argument for launching a study into dominions in quasivarieties of universal algebras, undertaken in [8]. There, the concept of a dominion is extended to the case A ∈ M, which turns out useful in dealing with dominions in quasivarieties.…”

“…is the lattice of subquasivarieties of a quasivariety M. Also, in [8], conditions were specified under which L(A, H, M) forms a lattice under set-theoretic inclusion, and the problem was posed as to the interplay between the lattices L q (M) and L(A, H, M). In particular, a question was dubbed asking which conditions are necessary for the map ϕ : …”