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A characterization of the nn-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)(A,J) and a natural number nn, the closure operator JJ on the set AA is nn-ary if, and only if, there exists a single-sorted signature Σ\Sigma and a Σ\Sigma-algebra A\mathbf{A} such that every operation of A\mathbf{A} is of an arity n\leq n and J=SgAJ = \mathrm{Sg}_{\mathbf{A}}, where SgA\mathrm{Sg}_{\mathbf{A}} is the subalgebra generating operator on AA determined by A\mathbf{A}. On the other hand, a theorem of Tarski asserts that if JJ is an nn-ary closure operator on a set AA with n2n\geq 2, and if i<ji<j with ii, jIrB(A,J)j\in \mathrm{IrB}(A,J), where IrB(A,J)\mathrm{IrB}(A,J) is the set of all natural numbers nn such that (A,J)(A,J) has an irredundant basis (\equiv minimal generating set) of nn elements, such that {i+1,,j1}IrB(A,J)=\{i+1,\ldots, j-1\}\cap \mathrm{IrB}(A,J) = \varnothing, then jin1j-i\leq 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

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