Epic substructures and primitive positive functions

Abstract

For $\mathbf{A}\le \mathbf{B}$ first order structures in a class $\mathcal{K}$, say that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in $\mathcal{K}$ if for every $\mathbf{C}\in \mathcal{K}$ and all homomorphisms $g,g^{\prime }:\mathbf{B}\to \mathbf{C}$, if $g$ and $g^{\prime }$ agree on $A$, then $g=g^{\prime }$. We prove that $\mathbf{A}$ is an epic substructure of $\mathbf{B}$ in a class $\mathcal{K}$ closed under ultraproducts if and only if $A$ generates $\mathbf{B}$ via operations definable in $\mathcal{K}$ with primitive positive formulas. Applying this result we show that a quasivariety of algebras $\mathcal{Q}$ with an $n$-ary near-unanimity term has surjective epimorphisms if and only if ${\mathbb{SP}}_n{\mathbb{P}}_u({\mathcal{Q}}_{RSI})$ has surjective epimorphisms. It follows that if $\mathcal{F}$ is a finite set of finite algebras with a common near-unanimity term, then it is decidable whether the (quasi)variety generated by $\mathcal{F}$ has surjective epimorphisms.

11 pages, 1 figure