# Epic substructures and primitive positive functions

11 Jul 2016 Campercholi Miguel

For $\mathbf{A}\leq\mathbf{B}$ 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 $g,g^{\prime}:\mathbf{B}\rightarrow\mathbf{C}$, if $g$ and $g'$ agree on $A$, then $g=g'$. 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... (read more)

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