Multi-argument specialization semilattices

If $X$ is a closure space with closure $K$, we consider the semilattice $(\mathcal P(X), \cup)$ endowed with a further relation $ x \sqsubseteq\{ y_1, y_2, \dots, y_n\} $ between elements of $\mathcal P(X)$ and finite subsets of $\mathcal P(X)$, whose interpretation is $x \subseteq Ky_1 \cup Ky_2 \cup\dots\cup Ky_n $. We present axioms for such multi-argument specialization semilattices and show that this list of axioms is sound and complete for substructures of closure spaces, namely, a model satisfies the axioms if and only if it can be embedded into the structure associated to a closure space as in the previous sentence. As a main tool for the proof, we provide a canonical embedding of a multi-argument specialization semilattice into (the structure associated to) a closure semilattice.