#### Presentation Title

Linear Systems and Boolean Algebra

#### Faculty Mentor

Dr. Jeremy Kepner

#### Start Date

18-11-2017 11:15 AM

#### End Date

18-11-2017 11:30 AM

#### Location

9-285

#### Session

Physical Sciences 2

#### Type of Presentation

Oral Talk

#### Subject Area

physical_mathematical_sciences

#### Abstract

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings 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.Here we compute the solution space of a linear system over an arbitrary complete Boolean algebra, expressed in the form of a union of closed intervals.

#### Summary of research results to be presented

The structure theorem indicates that the solution space of join-blank algebra is represented as a union of closed intervals of same terminal point. The added ability to take complements in a Boolean algebra allows for exact computation of the solution space, using the maximum solutions calculated with complements (exact equation needs a lot of formatting, which will be presented in a LaTeX environment). This can be further applied to find a solution set in a topology as a lattice.

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Linear Systems and Boolean Algebra

9-285

A central problem of linear algebra is solving linear systems. Regarding linear systems as equations over general semirings 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.Here we compute the solution space of a linear system over an arbitrary complete Boolean algebra, expressed in the form of a union of closed intervals.