Linear Systems over Join-Blank Algebras

TL;DR

This paper investigates solutions to linear systems over join-blank and max-blank algebras, showing that solution spaces can be characterized as finite unions of quasi-intervals, extending previous results.

Contribution

It generalizes the structure of solution spaces for linear systems over join-blank algebras, removing the need for the closed interval hypothesis.

Findings

Solution spaces are unions of closed intervals in join-blank algebras.

In max-blank algebras, solution spaces are finite unions of closed intervals under certain conditions.

Without the closed interval hypothesis, solution spaces are finite unions of quasi-intervals.

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.