An Extension Theorem for Signotopes

Abstract: In 1926, Levi showed that, for every pseudoline arrangement $A$ and two points in the plane, $A$ can be extended by a pseudoline which contains the two prescribed points. Later extendability was studied for arrangements of pseudohyperplanes in higher dimensions. While the extendability of an arrangement of proper hyperplanes in R^d with a hyperplane containing $d$ prescribed points is trivial, Richter-Gebert found an arrangement of pseudoplanes in R^3 which cannot be extended with a pseudoplane containing two particular prescribed points. In this talk, we investigate the extendability of signotopes, which are a combinatorial structure encoding a rich subclass of pseudohyperplane arrangements. We show that signotopes of odd rank are extendable in the sense that for two prescribed crossing points we can add an element containing them.

machine learningcommutative algebraalgebraic geometryalgebraic topologycombinatoricscategory theoryoperator algebrasrings and algebrasrepresentation theory

Audience: researchers in the topic

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Algebraic and Combinatorial Perspectives in the Mathematical Sciences

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