## A characterization of the $$n$$-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem.(English)Zbl 07144285

Summary: 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 $$\Sigma$$ and a $$\Sigma$$-algebra A such that every operation of A is of an arity $$\leq n$$ and $$J = \mathrm{Sg}\mathbf{A}$$, where $$\mathrm{Sg}\mathbf{A}$$ 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 $$n\geq 2$$, then, for every $$i, j \in \mathrm{IrB}(A,J)$$, where $$\mathrm{IrB}(A,J)$$ is the set of all natural numbers which have the property of being the cardinality of an irredundant basis ($$\equiv$$ minimal generating set) of $$A$$ with respect to $$J$$ , if $$i<j$$ and $$i+1\{i+1,\dots, j-1\}\cap\mathrm{IrB}(A,J)=\varnothing$$, then $$j-i\leq n-1$$. 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.

### MSC:

 06A15 Galois correspondences, closure operators (in relation to ordered sets) 54A05 Topological spaces and generalizations (closure spaces, etc.)
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### References:

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