A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem. Issue 10 (26th November 2019) - Climent Vidal, J.; Cosme Llópez, E.
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 = 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, 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 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.