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 Σ and a
Σ-algebra A such that every operation of A is of
an arity ≤n and J=SgA, where
SgA is the subalgebra generating operator on A
determined by 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≥2, and if i<j
with i, j∈IrB(A,J), where IrB(A,J) is the set of all
natural numbers n such that (A,J) has an irredundant basis (≡
minimal generating set) of n elements, such that {i+1,…,j−1}∩IrB(A,J)=∅, then j−i≤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