On the Possibilities of Defining Infinite Oriented Matroids

Abstract

Is it possible to define cryptomorphic axiom systems for infinite oriented matroids by lifting some of the axiom systems for finite oriented matroids to the infinite setting while not losing duality in the process? We show that the answer to this question is a twofold "no". First, lifting the circuit axioms neither preserves duality nor inheritance of strong circuit elimination in minors. Second, although duality is kept intact by translating the orthogonality axioms and an axiom system based on the Farkas Lemma, the classes of infinite oriented matroids obtained in this way have the property that one is a proper subclass of the other.

Figures (8)

• Figure 1: Five lines $a,b,c,d,e$ through the origin in $\mathbb{R}^3$ such that the intersection of the planes spanned by $ab$ (blue) and $cd$ (green) is a line through the origin
• Figure 2: If the lines $a,b,c,d,e$ from Figure \ref{['fig:NeatSetFiveLines']} belong to a neat set of lines, then there exists a further line $f$ such that the intersection of the planes spanned by $ab$ (blue), $cd$ (green), and $ef$ (orange) is a line again
• Figure 3: Signed circuits and cocircuits of $M$ from Example \ref{['exmp:CMWithoutFarkas']}
• Figure 4: Digraph of Example \ref{['exmp:NonConfCircAfterElim']}
• Figure 5: The Bean Graph

Theorems & Definitions (133)

• Definition 2.1
• Definition 2.2
• Theorem 2.3
• proof
• Definition 2.4
• Proposition 2.5
• proof
• Definition 2.6
• Example 2.7
• Example 2.8