Definability via partial enumerations with semicomputable codomains

Abstract

Let $\mathfrak{A}$ be a total abstract structure. We prove that if a set $A\subseteq |\mathfrak{A}{|}^n$ is admissible in every partial enumeration of $\mathfrak{A}$ with semicomputable codomain, then $A$ is semicomputable in $\mathfrak{A}$ in the sense of Friedman - Shepherdson.