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Published in:
2020 | OriginalPaper | Chapter
n-Ary k-Actions Between Sets and Their Applications
Authors : Antonio J. Calderón Martín, Babacar Gaye, Francisco J. Navarro Izquierdo
Published in: Associative and Non-Associative Algebras and Applications
Publisher: Springer International Publishing
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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.
$$f : \overset{k)}{\mathfrak A\times \cdots \times \mathfrak A} \times \overset{n-k)}{\mathfrak B\times \cdots \times \mathfrak B} \rightarrow \mathfrak A$$