Linear Systems over Join-Blank Algebras
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Publication:6292352
DOI10.1109/URTC.2017.8284192arXiv1710.03381MaRDI QIDQ6292352
Jeremy Kepner, Suna Kim, Hayden Jananthan
Publication date: 9 October 2017
Abstract: A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings (V,otimes,oplus,0,1) instead of rings or fields makes traditional approaches impossible. Earlier work shows that the solution space X(A;w) of the linear system Av = w over the class of semirings called join-blank algebras is a union of closed intervals (in the product order) with a common terminal point. In the smaller class of max-blank algebras, the additional hypothesis that the solution spaces of the 1x1 systems Av = w are closed intervals implies that X(A;w) is a finite union of closed intervals. We examine the general case, proving that without this additional hypothesis, we can still make X(A;w) into a finite union of quasi-intervals.
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