No abstract

No 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.

Let A be a universal algebra and H its subalgebra. The dominion of H in A (in a class M) is the set of all elements a ∈ A such that every pair of homomorphisms f, g : A → M ∈ M satisfies the following: if f and g coincide on H, then f (a) = g(a). A dominion is a closure operator on a set of subalgebras of a given algebra. The present account treats of closed subalgebras, i.e., those subalgebras H whose dominions coincide with H. We introduce projective properties of quasivarieties which are similar to the projective Beth properties dealt with in nonclassical logics, and provide a characterization of closed algebras in the language of the new properties. It is also proved that in every quasivariety of torsion-free nilpotent groups of class at most 2, a divisible Abelian subgroup H is closed in each group H, a generated by one element modulo H.

No abstract