A characterization of the $n$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem

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 $\mathrm{\Sigma }$ and a $\mathrm{\Sigma }$-algebra $\mathbf{A}$ such that every operation of $\mathbf{A}$ is of an arity $\le n$ and $J={\mathrm{S}\mathrm{g}}_{\mathbf{A}}$, where ${\mathrm{S}\mathrm{g}}_{\mathbf{A}}$ 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 $n\ge 2$, and if $i<j$ with $i$, $j\in \mathrm{I}\mathrm{r}\mathrm{B}(A,J)$, where $\mathrm{I}\mathrm{r}\mathrm{B}(A,J)$ 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 $\mathrm{I}\mathrm{r}\mathrm{B}$(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.