  A theorem of single-sorted algebra states that, for a closure space $(A,J)$
and a natural number $n$, the closure operator $J$ on the set $A$ is $n$-ary
if, and only if, there exists a single-sorted signature $\Sigma$ and a
$\Sigma$-algebra $\mathbf{A}$ such that every operation of $\mathbf{A}$ is of
an arity $\leq n$ and $J = \mathrm{Sg}_{\mathbf{A}}$, where
$\mathrm{Sg}_{\mathbf{A}}$ is the subalgebra generating operator on $A$
determined by $\mathbf{A}$. On the other hand, a theorem of Tarski asserts that
if $J$ is an $n$-ary closure operator on a set $A$ with $n\geq 2$, and if $i<j$
with $i$, $j\in \mathrm{IrB}(A,J)$, where $\mathrm{IrB}(A,J)$ is the set of all
natural numbers $n$ such that $(A,J)$ has an irredundant basis ($\equiv$
minimal generating set) of $n$ elements, such that $\{i+1,\ldots, j-1\}\cap
\mathrm{IrB}(A,J) = \varnothing$, then $j-i\leq n-1$. In this article we state
and prove the many-sorted counterparts of the above theorems. But, we remark,
regarding the first one under an additional condition: the uniformity of the
many-sorted closure operator.
