Linear systems over join-blank algebras

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings (V, ⊕, ⊗, 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 1 × 1 systems A ⊗ v = 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.