The set of all [group/ring/module/etc.] structures on a cardinal number Given a cardinal number $\kappa$, how can one construct the set of the (algebraic) structures of a certain kind (group, ring, module, etc.) whose base set is a subset of $\kappa$?
For example, using the axioms of ZFC, how can one justify the existence of the set of all group structures whose base set is a subset of the cardinal $\kappa$? What would this set look like with the set-builder notation?
 A: I'll address the case of groups first, since I imagine you'll probably be able to extend it readily to the other cases.
At the end of the day, a group is a set $G$ equipped with a choice of an element $e\in G$ and a choice of a function $\cdot:G\times G\to G$ satisfying some properties.
Therefore, for any set $G$, the set of all group structures on $G$ is a subset of $G\times(G^{G\times G})$.
Now, for a cardinal $\kappa$ (or any set, really), we have a set of subsets $G\subseteq\kappa$, for which we get a set of group structures contained in $G\times(G^{G\times G})$, so the set of all groups whose base set sits inside $\kappa$ is a subset of
$$
\bigcup_{G\subseteq\kappa}G\times(G^{G\times G})
$$
which, being a set-indexed union of sets, is a set.
If you wanted to describe this set explicitly using set-builder notation, you can just define it set $\{(G,e,\cdot) \mid G\subseteq\kappa; e\in G; (\cdot):G\times G\to G; \text{group axioms}\}$ or even
$$
\left\{(G,e,\cdot)\in\bigcup_{G\in2^\kappa}\{G\}\times G\times(G^{G\times G}) \middle| \text{group axioms}\right\}
$$
