A characterization of the n-ary many-sorted closure operators and a many-sorted Tarski irredundant basis theorem. Issue 10 (26th November 2019)

• Record Type: Journal Article
• Authors: Climent Vidal, J.; Cosme Llópez, E.
• Is Part Of: Quaestiones mathematicae. Volume 42:Issue 10(2019)
• Page Start: 1427
• Page End: 1444
• Publication Date: 2019-11-26
• Subjects: Primary: 06A15 -- Secondary: 54A05; 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; Mathematics -- Periodicals; 510.5
• DOI: 10.2989/16073606.2018.1532931
• Languages: English
• ISSNs: 1607-3606

Abstract

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.