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Let A ≤ B be structures, and 
${\cal K}$ a class of structures. An element b ∈ B is dominated by A relative to 
${\cal K}$ if for all 
${\bf{C}} \in {\cal K}$ and all homomorphisms g, g' : B → C such that g and g' agree on A, we have gb = g'b. Our main theorem states that if 
${\cal K}$ is closed under ultraproducts, then A dominates b relative to 
${\cal K}$ if and only if there is a partial function F definable by a primitive positive formula in 
${\cal K}$ such that FB(a1,…,an) = b for some a1,…,an ∈ A. Applying this result we show that a quasivariety of algebras 
${\cal Q}$ with an n-ary near-unanimity term has surjective epimorphisms if and only if 
$\mathbb{S}\mathbb{P}_n \mathbb{P}_u \left( {\mathcal{Q}_{{\text{RSI}}} } \right)$ has surjective epimorphisms. It follows that if 
${\cal F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by 
${\cal F}$ has surjective epimorphisms.
