Abstract

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