DOMINIONS AND PRIMITIVE POSITIVE FUNCTIONS
Miguel A. Campercholi · Journal of Symbolic Logic · 2018
Abstract LetA≤Bbe structures, and ${\cal K}$ a class of structures. An elementb∈BisdominatedbyArelative to ${\cal K}$ if for all ${\bf{C}} \in {\cal K}$ and all homomorphismsg,g':B → Csuch thatgandg'agree onA, we havegb=g'b. Our main theorem states that if ${\cal K}$ is closed under ultraproducts, thenAdominatesbrelative to ${\cal K}$ if and only if there is a partial functionFdefinable by a primitive positive formula in ${\cal K}$ such thatFB(a1,…,an) =bfor somea1,…,an∈A. Applying this result we show that a quasivariety of algebras ${\cal Q}$ with ann-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.