Abstract:
We fix a universal algebra $A$ and its subalgebra $H$. The dominion of $H$ in $A$ (in a class $\mathcal M$) is the set of all elements $a\in A$ such that any pair of homomorphisms $f,g:A\rightarrow M\in\mathcal M$ satisfies the following: if $f$ and $g$ coincide on $H$ then $f(a)=g(a)$. In association with every quasivariety, therefore, is a dominion of $H$ in $A$. Sufficient conditions are specified under which a set of dominions form a lattice. The lattice of dominions is explored for down-semidistributivity. We point out a class of algebras (including groups, rings) such that every quasivariety in this class contains an algebra whose lattice of dominions is anti-isomorphic to a lattice of subquasivarieties of that quasivariety.
Keywords:dominion, lattice of dominions, quasivariety.
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