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J. Climent Vidal, E. Cosme Llópez.
A characterization of the $n$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem.
Quaest. Math., 42:1427-1444, 2019.

doi: 10.2989/16073606.2018.1532931

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 ) = 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.

2020 Mathematics Subject Classification: 06A15, 54A05.

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