Definability via partial enumerations with semicomputable codomains (Annual of Sofia University St. Kliment Ohridski. Faculty of Mathematics and Informatics, 2000)
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.