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**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.) |

### 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
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\textit{J. Climent Vidal} and \textit{E. Cosme Llópez}, Quaest. Math. 42, No. 10, 1427--1444 (2019; Zbl 07144285)

### References:

[1] | Birkhoff, G.; Frink, O., Representation of lattices by sets, Trans. Amer. Math. Soc., 64, 299-316 (1948) · Zbl 0032.00504 |

[2] | Burris, S.; Sankappanavar, H. P., A course in universal algebra (1981), Springer-Verlag: Springer-Verlag, New York/Berlin · Zbl 0478.08001 |

[3] | Climent Vidal, J.; Soliveres Tur, J., On many-sorted algebraic closure operators, Math. Nachr., 266, 81-84 (2004) · Zbl 1038.08001 |

[4] | Tarski, A., An interpolation theorem for irredundant bases of closure operators, Discrete Math., 12, 185-192 (1975) · Zbl 0319.06002 |

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