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Erschienen in:
2020 | OriginalPaper | Buchkapitel
n-Ary k-Actions Between Sets and Their Applications
verfasst von : Antonio J. Calderón Martín, Babacar Gaye, Francisco J. Navarro Izquierdo
Erschienen in: Associative and Non-Associative Algebras and Applications
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Abstract
We consider families F of n-ary k-actions between arbitrary non-empty sets \(\mathfrak A\) and \(\mathfrak B\) and show that if every \(f\in F\) fixes some element in \(\mathfrak A\), then this family induces an adequate decomposition of \(\mathfrak A\) as the (orthogonal) disjoint-pointed union of well-described F-invariant subsets (F-submodules). If \(\mathfrak A\) is furthermore a division F-module, it is shown that the above decomposition is by means of the family of its pointed simple F-submodules. The obtained results are applied to the structure theory of arbitrarily graded n-linear k-modules by stating a second Wedderburn type theorem for the class of n-linear k-modules with an arbitrary division grading.
$$f : \overset{k)}{\mathfrak A\times \cdots \times \mathfrak A} \times \overset{n-k)}{\mathfrak B\times \cdots \times \mathfrak B} \rightarrow \mathfrak A$$