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

### Abstract

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 = Sg_{A}, where Sg_{A} 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, 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) = ∅, 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.

** Key words**: S-sorted set, delta of Kronecker, support of an S-sorted set, n-ary manysorted closure operator, uniform many-sorted closure operator, irredundant basis with respect to a many-sorted closure operator.