Conditional Oriented Matroids

• Conditional oriented matroids are sign-vector systems that generalize oriented matroids by retaining face symmetry and strong elimination without requiring a global zero covector.
• They are embodied in realizable models like restricted hyperplane arrangements, serving as a bridge between affine oriented matroids and lopsided systems.
• Their framework enables efficient computational reconstruction of covector sets, tope graph analysis, and extensions of determinantal formulas in generalized orientation theories.

Conditional oriented matroids (COMs) are sign-vector systems that generalize oriented matroids, affine oriented matroids, and lopsided sets by retaining face symmetry and strong elimination while not requiring the zero covector or global symmetry/composition. In the standard sign-vector language, a COM is a nonempty set $\mathcal L\subseteq\{+,-,0\}^E$ (equivalently $\{-1,0,1\}^E$) of covectors such that $X\circ -Y\in\mathcal L$ for all $X,Y\in\mathcal L$, and whenever $e\in Sep(X,Y)$, there exists $Z\in\mathcal L$ with $Z_e=0$ and $Z_f=(X\circ Y)_f$ for all $f\in E\setminus Sep(X,Y)$. If $\mathcal L$ also contains the zero vector, then it is an ordinary oriented matroid (Bandelt et al., 2015, Randriamaro, 2023).

1. Axiomatic framework and position among sign-vector systems

A COM is built from the same basic sign-vector operations as oriented matroid theory. For $\{-1,0,1\}^E$0, the composition is defined coordinatewise by

$\{-1,0,1\}^E$1

and the separator is

$\{-1,0,1\}^E$2

A partial order is defined by

$\{-1,0,1\}^E$3

so that $\{-1,0,1\}^E$4 is obtained from $\{-1,0,1\}^E$5 by replacing some nonzero entries by $\{-1,0,1\}^E$6. Maximal elements in this order are the topes (Randriamaro, 2023).

The defining COM axioms are face symmetry and strong elimination. Face symmetry is

$\{-1,0,1\}^E$7

and strong elimination requires that for each $\{-1,0,1\}^E$8 and each $\{-1,0,1\}^E$9, there exists $X\circ -Y\in\mathcal L$0 such that

$X\circ -Y\in\mathcal L$1

These axioms preserve the elimination-theoretic core of oriented matroids while weakening their global symmetry package (Bandelt et al., 2015).

Within the standard hierarchy of sign-vector systems, the principal classes are separated by four axioms: composition $X\circ -Y\in\mathcal L$2, strong elimination $X\circ -Y\in\mathcal L$3, symmetry $X\circ -Y\in\mathcal L$4, face symmetry $X\circ -Y\in\mathcal L$5, and ideal composition $X\circ -Y\in\mathcal L$6. A strong elimination system satisfies $X\circ -Y\in\mathcal L$7; a COM satisfies $X\circ -Y\in\mathcal L$8; an oriented matroid satisfies $X\circ -Y\in\mathcal L$9; and a lopsided system satisfies $X,Y\in\mathcal L$0. In this sense COMs sit between oriented matroids and lopsided systems (Bandelt et al., 2015).

A precise relation to oriented matroids is given by the zero-vector condition

$X,Y\in\mathcal L$1

An oriented matroid is exactly a COM satisfying this condition. Conversely, if a COM contains the zero vector, then face symmetry with $X,Y\in\mathcal L$2 yields symmetry, and composition follows from face symmetry applied to $X,Y\in\mathcal L$3 and $X,Y\in\mathcal L$4 (Randriamaro, 2023).

2. Realizable models and principal examples

The fundamental realizable model of a COM is a hyperplane arrangement restricted to an open convex set. Given a central or affine hyperplane arrangement in $X,Y\in\mathcal L$5 and an open convex set $X,Y\in\mathcal L$6, one restricts the arrangement sign vectors to the regions inside $X,Y\in\mathcal L$7; the resulting sign-vector system is a realizable COM. This construction unifies several familiar special cases: if the arrangement is central and $X,Y\in\mathcal L$8 contains the origin, one recovers an oriented matroid; if the arrangement is affine and $X,Y\in\mathcal L$9, one obtains a realizable affine oriented matroid; and for the coordinate arrangement one gets realizable lopsided sets (Bandelt et al., 2015).

A more explicit arrangement-pair model uses a cooriented affine arrangement $e\in Sep(X,Y)$0 in a real vector space $e\in Sep(X,Y)$1 together with a nonempty convex open set $e\in Sep(X,Y)$2. For each sign vector $e\in Sep(X,Y)$3, define

$e\in Sep(X,Y)$4

Then

$e\in Sep(X,Y)$5

is a conditional oriented matroid. When $e\in Sep(X,Y)$6, this is an oriented matroid exactly when the hyperplanes have a common intersection point in $e\in Sep(X,Y)$7 (Dorpalen-Barry et al., 8 Jul 2025).

Apartments of hyperplane arrangements form another important class of realizable COMs. If $e\in Sep(X,Y)$8 is a chamber of a subarrangement, then after deleting the hyperplanes disjoint from $e\in Sep(X,Y)$9, the induced sign system is a conditional oriented matroid. This realization is central to the use of COMs in computation, because an apartment need not carry the symmetry and zero-vector structure of an oriented matroid while still admitting a valid COM encoding (Randriamaro, 2023).

The example theory extends beyond arrangements. For a poset $Z\in\mathcal L$0, the ranking COM $Z\in\mathcal L$1 is a realizable COM whose topes are exactly the linear extensions of the poset and whose tope graph is the linear extension graph. More generally, relaxing realizability to local realizability yields COMs from non-positively curved Coxeter zonotopal complexes; in that setting, the resulting sign-vector system is a simple COM whose tope graph is the graph of the zonotopal complex (Bandelt et al., 2015).

3. Faces, topes, cocircuits, and decomposition into oriented-matroid pieces

For a covector $Z\in\mathcal L$2, the standard notation is

$Z\in\mathcal L$3

The face of $Z\in\mathcal L$4 is

$Z\in\mathcal L$5

If $Z\in\mathcal L$6, then $Z\in\mathcal L$7; otherwise,

$Z\in\mathcal L$8

Closure under deletion and contraction holds, and every fiber of a COM is again a COM. Moreover, if $Z\in\mathcal L$9 is fixed, then the face $Z_e=0$0 yields an oriented matroid after contraction: $Z_e=0$1 This is the local mechanism behind the phrase “complexes of oriented matroids” (Bandelt et al., 2015).

Topes are the maximal sign vectors in the natural partial order. For semisimple COMs, the set of topes determines the COM uniquely, and the tope graph is a partial cube. In fact, a semisimple strong elimination system has a partial-cube tope graph, with edges corresponding to sign vectors having singleton zero sets; a simple COM is determined by its tope graph up to reorientation (Bandelt et al., 2015). In a broader graph-theoretic characterization, COMs correspond to partial cubes whose antipodal subgraphs are gated (Knauer et al., 2020).

The cocircuit theory of COMs parallels the oriented-matroid cocircuit formalism but is intrinsically one-sided. Let $Z_e=0$2 denote the supremum-irreducible sign vectors. A minimal sign vector of $Z_e=0$3 is called an improper cocircuit, and a proper cocircuit is a sign vector covering an improper cocircuit. The cocircuit set $Z_e=0$4 then consists of the improper and proper cocircuits together, and it characterizes the COM via the axioms $Z_e=0$5, $Z_e=0$6, and $Z_e=0$7 (Bandelt et al., 2015).

Structurally, every semisimple COM can be built by successive COM amalgamations from its maximal faces, and those maximal faces contract to oriented matroids. This gives a binary composition scheme in which the global COM is assembled from oriented-matroid pieces. The associated Euler–Poincaré relation is

$Z_e=0$8

For lopsided systems, where $Z_e=0$9, this becomes the zero-set Euler formula

$Z_f=(X\circ Y)_f$0

These identities distinguish COMs from arbitrary sign systems and identify lopsided systems inside the COM class (Bandelt et al., 2015).

4. Reconstruction from topes and computational use

A major reconstruction theorem for COMs generalizes Mandel’s theorem for oriented matroids. If $Z_f=(X\circ Y)_f$1 is a conditional oriented matroid with tope set $Z_f=(X\circ Y)_f$2, then

$Z_f=(X\circ Y)_f$3

Thus the full covector set is determined by the topes alone, but in the COM setting the criterion uses $Z_f=(X\circ Y)_f$4, not $Z_f=(X\circ Y)_f$5 as in the oriented-matroid case (Randriamaro, 2023).

The proof mechanism reflects the COM axioms rather than oriented-matroid composition. Forward inclusion uses face symmetry: if $Z_f=(X\circ Y)_f$6 and $Z_f=(X\circ Y)_f$7 is a tope, then $Z_f=(X\circ Y)_f$8, and maximality forces it to be a tope. Reverse inclusion proceeds by induction on $Z_f=(X\circ Y)_f$9 and $f\in E\setminus Sep(X,Y)$0, combining deletion, contraction, adjacency of topes differing in one coordinate, and strong elimination (Randriamaro, 2023).

This theorem has direct algorithmic consequences. The procedure GeneratingCOM(T) takes a tope set $f\in E\setminus Sep(X,Y)$1, enumerates all $f\in E\setminus Sep(X,Y)$2, and keeps precisely those satisfying

$f\in E\setminus Sep(X,Y)$3

A second procedure, ApartmentToCOM(A,P), starts from a hyperplane arrangement and a set of sample points chosen inside chambers, computes the resulting topes, and then applies GeneratingCOM. These algorithms are presented as practical tools for converting apartments into COMs (Randriamaro, 2023).

The computational significance is explicit: COMs are described as more suitable for computer calculations than ordinary oriented matroids for certain geometric configurations, especially apartments of hyperplane arrangements. The cited applications include determining the covector set, computing the $f\in E\setminus Sep(X,Y)$4-polynomial, and computing other invariants like the Varchenko determinant after converting an apartment into a COM. In the worked example from a specific arrangement in $f\in E\setminus Sep(X,Y)$5, the apartment $f\in E\setminus Sep(X,Y)$6-polynomial is

$f\in E\setminus Sep(X,Y)$7

This suggests a computational workflow in which topes are primary data and the covector structure is reconstructed afterward (Randriamaro, 2023).

5. Topology, Salvetti complexes, and tope-graph geometry

COMs admit a topological representation by regular cell complexes. For a COM $f\in E\setminus Sep(X,Y)$8, there is a regular cell complex $f\in E\setminus Sep(X,Y)$9 that is contractible, and the tope graph of $\mathcal L$0 is the $\mathcal L$1-skeleton of $\mathcal L$2. Realizable COMs are zonotopally realizable, hence locally realizable, and each face is combinatorially a zonotope (Bandelt et al., 2015).

A stronger arrangement-theoretic model is provided by the Salvetti construction in the conditional setting. For a COM $\mathcal L$3, the Salvetti poset consists of pairs $\mathcal L$4 with $\mathcal L$5 and $\mathcal L$6 a tope, ordered by

$\mathcal L$7

For an arrangement pair $\mathcal L$8, the order complex $\mathcal L$9 is homotopy equivalent to the partial complexified complement $\{-1,0,1\}^E$00, and also to the canonical non-Hausdorff gluing space $\{-1,0,1\}^E$01: $\{-1,0,1\}^E$02 This extends the classical Salvetti theorem from oriented matroids to conditional oriented matroids (Dorpalen-Barry et al., 8 Jul 2025).

The proof uses a nerve-style argument adapted to a cover that is not necessarily a good cover. For each $\{-1,0,1\}^E$03, one defines a contractible set $\{-1,0,1\}^E$04, and for each point $\{-1,0,1\}^E$05 the local poset

$\{-1,0,1\}^E$06

is shown to be isomorphic to a covector poset $\{-1,0,1\}^E$07. Since $\{-1,0,1\}^E$08 is contractible, the local order complexes are contractible, and the resulting homotopy-colimit argument yields $\{-1,0,1\}^E$09 (Dorpalen-Barry et al., 8 Jul 2025).

The geometry of tope graphs also supports a theory of corners. In the COM setting, corners generalize corners in lopsided sets and simplicial topes in oriented matroids. Realizable COMs, rank $\{-1,0,1\}^E$10 COMs, and hypercellular graphs admit corner peelings. This identifies broad classes in which the tope graph can be recursively decomposed while staying inside the COM category (Knauer et al., 2020).

6. Algebraic invariants, determinant phenomena, and broader conditionality

COMs support ring-theoretic invariants modeled on the classical Varchenko–Gelfand and Gelfand–Rybnikov constructions. If $\{-1,0,1\}^E$11 is the set of topes of a COM $\{-1,0,1\}^E$12, the Gelfand–Rybnikov ring is

$\{-1,0,1\}^E$13

For each ground-set element $\{-1,0,1\}^E$14, the Heaviside generators are

$\{-1,0,1\}^E$15

These generate $\{-1,0,1\}^E$16, and polynomial degree defines the Heaviside filtration. When $\{-1,0,1\}^E$17, this ring is exactly the Varchenko–Gelfand ring $\{-1,0,1\}^E$18 as a filtered ring (Dorpalen-Barry et al., 2022).

The same paper gives a cohomological interpretation of these constructions. For the arrangement pair $\{-1,0,1\}^E$19, there are canonical isomorphisms

$\{-1,0,1\}^E$20

$\{-1,0,1\}^E$21

$\{-1,0,1\}^E$22

with $\{-1,0,1\}^E$23. In the purely combinatorial direction, the ring, its associated graded, and its Rees algebra admit circuit-based presentations for arbitrary COMs (Dorpalen-Barry et al., 2022).

Determinant theory for COMs is less complete, but oriented-matroid results indicate a plausible extension pattern. The Varchenko matrix of an oriented matroid is defined directly from topes and separators,

$\{-1,0,1\}^E$24

and its determinant factors over covectors. The proof of the supertope contractibility theorem in that setting uses only covector elimination, and the authors explicitly remark that parts of the argument should extend beyond oriented matroids to COMs, particularly to structures such as closed supertopes without boundary. A plausible implication is that determinant formulas for broader sign-vector systems will depend on having suitable contraction/restriction behavior, a tope-poset topology, and a replacement for the contractibility of supertopes (Hochstättler et al., 2018).

COMs also belong to a wider relaxation program in matroidal sign theories. “Modular elimination in matroids and oriented matroids” shows that the oriented elimination axiom need only be imposed on modular pairs of signed circuits, so full elimination is redundant at the level of oriented-matroid circuit axioms (Delucchi, 2010). In a different direction, “Foundations for a theory of complex matroids” replaces signs by phases in $\{-1,0,1\}^E$25 and adopts modular elimination because general circuit elimination fails, even for realizable complex matroids (Anderson et al., 2010). These developments do not define COMs, but they clarify the role of conditional or restricted elimination in generalized orientation theories.

Taken together, these results place conditional oriented matroids at a junction of sign-vector combinatorics, hyperplane arrangement topology, partial-cube geometry, and filtered algebra. Their distinctive feature is that they preserve local oriented-matroid behavior—especially elimination and face structure—while allowing global asymmetry and the absence of the zero covector. That combination explains both their geometric origin in arrangements restricted to convex regions and their usefulness as a framework for extending oriented-matroid constructions beyond the classical symmetric setting (Bandelt et al., 2015).