The dominion of a subalgebra
H in an universal algebra
A (in a class
M\mathcal{M}
) is the set of all elements
a Î Aa \in A
such that for all homomorphisms
f,g:A ® B Î Mf,g:A \to B \in \mathcal{M}
if
f, g coincide on
H, then a
f = a
g. We investigate the connection between dominions and quasivarieties. We show that if a class
M\mathcal{M}
is closed under ultraproducts, then the dominion in
M\mathcal{M}
is equal to the dominion in a quasivariety generated by
M\mathcal{M}
. Also we find conditions when dominions in a universal algebra form a lattice and study this lattice.
Keywords Quasivariety - dominion - universal algebra - group - lattice - free amalgamated product - amalgam
Special issue of Studia Logica: 
Algebraic Theory of Quasivarieties
Presented by
M. E. Adams, K. V. Adaricheva, W. Dziobiak, and A. V. Kravchenko