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Galois connection for multiple-output operations. (English) Zbl 1402.08002

The paper presents a generalization of a special type of Galois connection, namely, given a base set \(B\), instead of classes of operations \(f:B^n\longrightarrow B\), the partial, multi-valued functions \(f:B^n\longrightarrow B^m\), \(m\geq 0\) are considered. Functions \(f:B^k\longrightarrow M\) valued in partially ordered monoids \(M\) as invariants are also used. As main results the paper presents:
(1)
the main Galois connection between partial multi-valued multi-output functions and pomonoid-valued weight functions in its most general form;
(2)
variants of Galois connections for restricted classes of multi-output functions or weights;
(3)
Galois connections for classes of total multi-output functions;
(4)
a description of finitely generated subdirectly irreducible commutative monoids.

MSC:

08A40 Operations and polynomials in algebraic structures, primal algebras
06F05 Ordered semigroups and monoids
06A15 Galois correspondences, closure operators (in relation to ordered sets)
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