
A theorem of singlesorted algebra states that, for a closure space (A, J ) and a natural number n, the closure operator J on the set A is nary if and only if there exists a singlesorted signature Σ and a Σalgebra 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 nary closure operator on a set A with n ≥ 2, then, for every i, j ∈ IrB(A, J ), where IrB(A, J ) is the set of all natural numbers which have the property of being the cardinality of an irredundant basis (≡ minimal generating set) of A with respect to J , if i < j and {i + 1, . . . , j − 1} ∩ IrB(A, J ) = Ø, then j − i ≤ n − 1. In this article we state and prove the manysorted counterparts of the above theorems. But, we remark, regarding the first one under an additional condition: the uniformity of the manysorted closure operator.
