Finitary affine oriented matroids

We initiate the axiomatic study of affine oriented matroids (AOMs) on arbitrary ground sets, obtaining fundamental notions such as minors, reorientations and a natural embedding into the frame work of Complexes of Oriented Matroids. The restriction to the finitary case (FAOMs) allows us to study tope graphs and covector posets, as well as to view FAOMs as oriented finitary semimatroids. We show shellability of FAOMs and single out the FAOMs that are affinely homeomorphic to $\mathbb{R}^n$. Finally, we study group actions on AOMs, whose quotients in the case of FAOMs are a stepping stone towards a general theory of affine and toric pseudoarrangements. Our results include applications of the multiplicity Tutte polynomial of group actions of semimatroids, generalizing enumerative properties of toric arrangements to a combinatorially defined class of arrangements of submanifolds. This answers partially a question by Ehrenborg and Readdy.

1. Introduction 1.1.Subject, results and structure of the paper.In this paper we establish the natural generalization of finite affine oriented matroids to arbitrary ground sets and derive several results about their axiomatics, topology and geometry.Our motivation is twofold: on the one hand we aim at advancing the structural theory of oriented matroids and arithmetic matroids, on the other hand we have in mind applications to linear and toric arrangements, which we discuss below in §1.2, as well as to general manifold arrangements (see Remark 1.1).Let us here summarize our main results.
‚ We present axiom systems for covectors of Affine Oriented Matroids (AOMs) over arbitrary ground sets (Section 2).These support canonical operations such as reorientation and taking minors ( §2.1).In particular our axiomatization, derived from results of Baum and Zhu [6], allows us to see AOMs as part of the theory of Complexes of Oriented Matroids (COMs) -a recent common generalization of oriented matroids and lopsided sets [4].However, the extension to arbitrary non-finite ground sets is novel and many of our results extend to general COMs, shedding a first light into this direction.Furthermore, we introduce a natural and geometrically meaningful notion of parallelism that defines an equivalence relation on the elements of the AOM ( §2.2).
It is crucial for the development of the subsequent results.‚ In order to obtain a theory that more closely encapsulates some of the geometric features of finitary affine hyperplane arrangements, in Section 3 we state axioms for Finitary Affine Oriented Matroids (FAOMs).These are AOMs with some local cardinality restrictions.A main theoretical feature of this restricted setting is that FAOMs are "orientations of finitary semimatroids", i.e.: the zero sets of covectors of an FAOM constitute the geometric semilattice of flats of a finitary semimatroid (e.g., in the sense of [17], generalizing the finite notion developed by Wachs and Walker [33] and by Ardila [3] and Kawahara [24]).We carry out a basic study of tope graphs and covector posets of FAOMs ( §3.1) and then we focus on topological properties.We prove that order complexes of covector posets of FAOMs are shellable ( §3.2) and explicitly describe their homeomorphism type ( §3.3).Moreover, we derive some order-theoretic properties of the geometric parallelism relation in FAOMs ( §3. 6).This allows us to single out a special class of FAOMs whose covector poset is affinely homeomorphic to Euclidean space R n (see Section 4).‚ In Section 5 we take FAOMs as a stepping stone in order to extend the theory of arrangements of pseudospheres (and -planes) beyond the Euclidean setting, towards pseudoarrangements in the torus.See §1.2 below for some motivating context from arrangements theory.In order to accomplish this we study group actions on AOMs and, in particular, a class of group actions for which the quotient of the covector poset is homeomorphic to a torus.In this torus, the quotients of all one-element contractions of the given FAOM determine an arrangement of tamely embedded tori.Notice that such "toric pseudoarrangements" are strictly more general than toric arrangements defined by level sets of characters (which we call "stretchable" extrapolating the Euclidean terminology), see Figure 1.2.1.In any case, stretchable or not, the faces of the corresponding dissection of the torus are enumerated by the Tutte polynomial associated in [17] to the induced group action on the underlying semimatroid, generalizing enumerative results by Moci on arithmetic Tutte polynomials associated to toric arrangements, see [12] and Remark A.17.We also mention that Pagaria in [30] put forward a notion of orientable arithmetic matroid, asking for an interpretation in terms of pseudoarrangements on the torus.See §6.2 for how our work contributes to this line of research.
Remark 1.1.Ehrenborg and Readdy ask in [18] for a natural class of submanifold arrangements where an "arithmetic" Tutte polynomial can be meaningfully defined.
Our first answer to this question is the class of arrangements in Euclidean space or in tori obtained from (possibly trivial) "sliding" group actions on FAOMs (Definition 5.1).Theorem 5. 22 shows that the Tutte polynomial of the associated group action on the underlying semimatroid provides the desired topological enumeration, together with the algebraic-combinatorial properties studied in [17].In the case of "standard" toric arrangements we recover the arithmetic Tutte polynomial.
The multi-pronged nature of our foundational work shows that infinite affine oriented matroids are at the crossroads of several topics in structural, algebraic and topological combinatorics.Thus AOMs offer new tools for existing open problems, and create some new ones in their own right: we outline some of these connections and research directions in Section 6.
In order to make the paper reasonably self-contained we include an Appendix where we briefly summarize the topological and algebraic-combinatorial tools we need.
1.2.Two motivating examples.We outline some of the motivation for our work, and explain our contribution in these contexts.
1.2.1.Arrangements in Euclidean space.Let A :" tH e u ePE be an arrangement of hyperplanes, i.e., a family of codimension 1 affine subspaces of the Euclidean space R d .We call such an arrangement "oriented" if for every e P E we are given a labeling by H è and H é of the two connected components of R d zH e .Definition 1.2.Given an oriented arrangement A :" tH e u ePE of affine hyperplanes in R d define, for every x P R d a sign vector Σ x P t`, 0, ´uE as follows.Σ x peq :" The covector axioms of oriented matroids abstract some of the properties of L pA q in the case where A is finite and XA ‰ H, see §A.3.Conversely, while not every oriented matroid arises from such an arrangement of hyperplanes, the powerful "Topological Representation Theorem" of Folkman and Lawrence asserts that the system of covectors of every oriented matroid is the set of sign vectors determined by some arrangement of oriented pseudospheres in the sphere (obtained as the boundary of the order complex of the covector poset, see [8,Chapter 5]).
If A is finite, but XA is not necessarily non-empty, then L pA q is the set of covectors of a finite affine oriented matroid.Finite affine oriented matroids can be defined either intrinsically or as subsets of covector sets of oriented matroids, see [6,23].The latter point of view allows us to interpret every finite affine oriented matroid as an arrangement of pseudoplanes in Euclidean space, again via the order complex of its covector poset, but it is an open problem to characterize which arrangements arise from finite affine oriented matroids, see [19] and §6.1 .
Figure 2. A non-stretchable line arrangement with an action of Z 2 defined by letting a lattice basis act as translations by the two sides of the shaded rectangle.(The picture should be thought of as being repeated in vertical and horizontal direction).Any orientation of it gives rise to a FAOM.More generally, if A is only assumed to be finitary, meaning that every x P R d has a neighborhood meeting finitely many H e , then every element of L pA q indexes an open cell in R d .These open cells are the relative interiors of the faces of the polyhedral subdivision of R d induced by A .The faces of a polyhedral complex are naturally ordered by inclusion, and this partial order corresponds to the (abstract) natural order among sign vectors (see Definition 2.1).
‚ Our "Finitary Affine Oriented Matroids" axiomatize properties of the polyhedral stratification of Euclidean space induced by finitary hyperplane arrangements.Not every FAOM is realizable as L pA q for a finitary arrangement.Still, some familiar geometric and topological features generalize nicely to the non-realizable case as well.
‚ We view our topological representation of FAOMs as a step towards the currently open problem of a topological characterization of affine pseudoarrangements (see §6.1).
1.2.2.Toric arrangements.Let now A be a finite family of level sets of characters of the compact torus T " pS1 q d .Such toric arrangements have been in the focus of recent research originally motivated by work of De Concini, Procesi and Vergne on partition functions and splines, see [15].A toric arrangement defines a polyhedral CW-structure KpA q on the torus.The face category of this cell complex is central in the study of the topology of the associated arrangement in the complex torus [14, §2] and of arrangements in products of elliptic curves [16].It can be regarded as the "toric" counterpart of the poset of faces of a linear arrangement 1 .Notice that, by passing to the universal cover of the torus, a toric arrangement can be seen as a quotient of an infinite, periodic arrangement of hyperplanes by the action of the deck transformation group.The current impulse towards the combinatorial study of toric arrangements already led to substantial algebraic-combinatorial developments such as arithmetic Tutte polynomials and arithmetic matroids [12].However, the only available results about the structure of face categories to date are an explicit description in the case of toric Weyl arrangements by means of "labelled necklaces" [1].
‚ We obtain an abstract characterization of the face category of toric arrangements as the quotient of the poset of covectors of an affine, infinite oriented matroid by a suitable class of group actions.This can be seen as an "oriented" version of the theory of group actions on semimatroids [17] designed to describe toric arrangements on an "unoriented", matroidal level.
‚ Accordingly, we obtain a notion of pseudoarrangements in the torus whose topology and geometry is amenable to treatment via the existing combinatorial toolkit.We leave the relationship to Pagaria's orientable arithmetic matroids to future research, see §6.

Affine oriented matroids (AOM)
In the non-finite context it is essential to view AOMs with an intrinsic axiomatization instead of as halfspaces of oriented matroids as described in §1.2.1.The goal of this section is to present the covector axiomatization of finite AOMs due to Karlander [23] whose proof was corrected recently by Baum and Zhu [6].We state this axiomatization for arbitrary cardinalities and bring it into a simplified form, which puts AOMs into the context of (complexes) of oriented matroids, (C)OMs [4].Moreover, we show that notions of minors and parallelism generalize straightforwardly to the infinite setting.Indeed, for the purpose of the present section no assumption on the ground set E has to be made.Definition 2.1.A sign vector (on a set E) is an element of t`, ´, 0u E .A system of sign vectors is any subset L Ď t`, ´, 0u E .We say system of sign vectors "on E", and write pE, L q, if specification is needed.Every system of sign vectors carries a natural partial order: X ď Y if and only if Xpeq ď Y peq for all e P E where we define 0 ă `, 0 ă ´, `and ´incomparable.The poset pL , ďq will be denoted by F pL q.
We introduce some further standard notions, see e.g.[8].The support of a sign vector X is X :" te P E | Xpeq ‰ 0u.The zero set of a sign vector X is the complement of its support, i.e., zepXq :" te P E | Xpeq " 0u.Moreover, the separator of two sign vectors X, Y is SpX, Y q :" te P X X Y | Xpeq ‰ Y pequ and the composition of X and Y is the sign vector given by X ˝Y peq :"

"
Xpeq if e P X Y peq otherwise.for all e P E.
We now recall some notations that we take from the specific treatment of the affine case given in [6].Let X, Y be any two sign vectors on E, e P E, and L a given system of sign vectors on E. Define
Moreover, set I " e pX, Y ; L q and IpX, Y ; L q :" ď ePSpX,Y q I e pX, Y ; L q.
We will omit reference to L , writing simply IpX, Y q, I " pX, Y q, etc., if no confusion can arise.The letter I is established for the above sets, because these sets can be seen as intervals of sorts, see Figure 4. Furthermore, write AsympL q :" tX P L | ´X R L u and let X ' Y be the sign vector obtained from "adding" the signs of the sum of X and Y seen as integer vectors, i.e., X ' Y peq :" for all e P E.
We can now set P " asym pL q :" tX ' p´Y q | X, Y P AsympL q, X " Y , I " pX, ´Y ; L q " I " p´X, Y ; L q " Hu, PpL q :" tX ' p´Y q | X, Y P L , IpX, ´Y ; L q " Ip´X, Y ; L q " Hu.
We are now able to state the first definition.
Definition 2.2 (AOM, following [6,23]).A pair pE, L q is the system of covectors of an affine oriented matroid if and only if (C) L ˝L Ď L , (composition) (FS) L ˝p´L q Ď L , (face symmetry) (SE " ) X, Y P L , X " Y ùñ @e P SpX, Y q : I " e pX, Y q ‰ H, (strong elimination equal support) (P " asym ) P " asym pL q ˝L Ď L .
(peripheral composition equal support) Then, the associated F pL q is called the poset of covectors of the given AOM.
We propose the following simpler and (seemingly) stronger axiomatization.
Proposition 2.4 (AOM).A pair pE, L q is the system of covectors of an affine oriented matroid if and only if (FS) L ˝p´L q Ď L , (SE) X, Y P L ùñ @e P SpX, Y q : I e pX, Y q ‰ H, (strong elimination) (P) PpL q ˝L Ď L .
We now move to prove the stated equivalence, in several steps.
Proof.It is enough to notice that X ˝Y " pX ˝´Xq ˝Y " X ˝p´X ˝Y q " X ˝´pX ˝´Y q. -(SE)ô(SE " ).
Proof.Clearly, (SE) implies (SE " ).Conversely, we get that with (C) the axiom (SE " ) implies (SE).Indeed, for X, Y P L by 1 we have I e pX, Y q " I e pX ˝Y, Y ˝Xq and both sets are defined for the same set of elements e.By ( 2) and (3) we have X ˝Y pf q " pX ˝Y q˝pY ˝Xqpf q, which gives I e pX ˝Y, Y ˝Xq " I " e pX ˝Y, Y ˝Xq.Thus, @e P SpX ˝Y, Y ˝Xq : I " e pX ˝Y, Y ˝Xq ‰ H implies @e P SpX, Y q : I e pX, Y q ‰ H. -(P)ñ(P " asym ).Proof.Since by ( 3) I e pX, Y q " I " e pX, Y q for sign vectors of equal support, we can write P " asym pL q as tX ' p´Y q | X, Y P AsympL q, X " Y , IpX, ´Y q " Ip´X, Y q " Hu, which gives P " asym pL q Ď PpL q. -Under (SE), (P " asym ) ñ (P).Proof.First observe that (P " asym ) ùñ tX ' p´Y q | X, Y P L , X " Y , IpX, ´Y q " Ip´X, Y q " Hu ˝L Ď L , i.e., we can drop the asymmetry condition.Indeed, suppose X, ´X, Y P L .Now, by (SE) Ip´X, Y q ‰ H except if Sp´X, Y q " H.But if X " Y then Sp´X, Y q " H implies Y " ´X, and X ' p´Y q " X ' X " X P L .Thus, only trivially fulfilled conditions are added.The symmetric argument works for the case X, Y, ´Y P L .
We proceed by showing that tX ' p´Y q | X, Y P L , X " Y , IpX, ´Y q " Ip´X, Y q " Hu Ě PpL q.
Let X ' p´Y q P PpL q and consider the vectors X ˝p´Y q and Y ˝p´Xq.We can compute ´pY ˝p´Xqq " ´Y ˝X and " ´Y ˝X " ´pY ˝p´Xqq " Y ˝p´Xq, where the last equality follows from Z " ´Z.By (1) we get SpX, ´Y q " SpX ˝p´Y q, ´Y ˝Xq and Sp´X, Y q " Sp´X ˝Y, Y ˝p´Xqq.

Which implies the equality of the elimination sets
IpX, ´Y q " IpX ˝p´Y q, ´Y ˝Xq and Ip´X, Y q " Ip´X ˝Y, Y ˝p´Xqq.
Finally, (4) gives X ' p´Y q " pX ˝p´Y qq ' p´Y ˝Xq.Together we obtain that X ' p´Y q is contained in the set on the left-hand side.This concludes the proof.

□
Example 2.5.In Figure 4 we illustrate the operations involved in the covector axioms of AOMs in Proposition 2.4 on the example of the realizable AOM from Figure 1.First, we choose two points X, Y identified with the corresponding sign vectors and add the auxiliary (dashed) line ℓ defined by the two points.The fact that a point on ℓ close to X towards Y as well as away from Y is also in the arrangement, respectively, corresponds to the covectors X ˝Y P L and X ˝´Y P L , respectively.The intersection points of ℓ with the hyperplanes H 2 and H 3 , respectively, correspond to the elements of IpX, Y ; L q. Now, Z, W are sign vectors corresponding to two maximal cells bounded by the parallel hyperplanes H 3 , H 4 .The sign vector Z ' ´W can be seen as the intersection point of the hyperplanes at infinity, i.e., with the auxiliary equator.The fact that pZ ' ´W q ˝L Ď L can be interpreted by saying that all points "close to pZ ' ´W q" towards existing cells of the arrangement also form part of the arrangement.Hence, the name peripheral composition.
Let us compare the axiomatization from Proposition 2.4 with axiomatizations for COMs and OMs, even if those are usually given for finite ground sets.Following [4] a COM is a system of sign vectors satisfying (FS) and (SE).Thus, from Proposition 2.4 one immediately deduces: Corollary 2.6.Every AOM is a COM.
The axiomatization of OMs given in [4] (see also Definition A.18) is as a system of sign vectors satisfying (FS) and (SE) and (0) the all-zeroes vector 0 is in L . (zero-vector) Corollary 2.7.Every OM is an AOM.
Proof.Note that (0) together with (FS) implies X P L ùñ ´X P L .This, together with (SE), yields that IpX, ´Y ; L q " SpX, ´Y q for all X, Y P L .Hence, PpL q Ď L ˝´L Ď L .Since (FS) implies L ˝L Ď L , (P) is fulfilled trivially .□ We now proceed to define reorientations.
Definition 2.8.Let L be a family of sign vectors.A reorientation of L is any set for a given τ P t`1, ´1u E , where multiplication is intended componentwise, i.e., pτ ¨Xqpeq :" τ peqXpeq.
Remark 2.9.It is straightforward to see that every reorientation of an AOM is an AOM.
2.1.Minors.The notion of minors is crucial in the study of OMs and COMs.Let us define the necessary ingredients here.Let pE, L q be any system of sign vectors.
Definition 2.10.For any A Ď E define the contraction of A in L as (notice that this is nonempty if and only if A P KpL q), and the deletion of A from L as Moreover, we call restriction to A the set L rAs :" L zpEzAq.
A system of sign vectors pE 1 , L 1 q is a minor of another system of sign vectors pE, L q if there are disjoint sets A, B Ď E such that pE 1 , L 1 q " pEzAzB, L zA{Bq.
As an example consider the AOM in Figure 1.Contracting the element 3 yields tp´, `, `q, p0, 0, `q, p`, ´, `qu.Thus can be seen as considering only the cells on H 3 and removing the third coordinate.Deleting 3, corresponds to removing H 3 from the arrangement.
Remark 2.11.Notice that there is a canonical order preserving injection Remark 2.12.One can show following [4], that for countable families tA i u iě1 and tB i u iě1 of sets we have i.e., the operations of contraction and deletion commute.We do not investigate further the case of uncountable families of sets.
Lemma 2.13.Let pE, L q satisfy (SE) and let A Ď E.Then, Proof.Let X ' ´Y P PpL zAq, with X, Y P L zA and I L zA pX, ´Y q " I L zA pX, ´Y q " H.We prove that there are X, Ŷ P L such that XzA " X, Ŷ zA " Y and X ' ´Ŷ P PpL q.Let X, Ŷ P L such that XzA " X, Ŷ zA " Y and suppose that X ' ´Ŷ R PpL q.Thus, without loss of generality there is a Z P I L p X, ´Ŷ q.
Note that for f P Sp X, ´Ŷ qzA we have Zpf q ‰ 0, since otherwise ZzA P I L zA pX, ´Y q with respect to f .Hence, zepZq " A Y pzepXq X zepY qq.
Furthermore note that Zpf q " Xf , since otherwise if Zpf q " ´Xpf q we can apply strong elimination to Z and X with respect to f and obtain a r Z P I L p X, ´Ŷ q with r Zpf q " 0. This contradicts the above.
We conclude that for all g P E we have Zpgq " Next, we show that Z ' ´Ŷ P PpL q.Clearly, we have Z, ´Ŷ P L .Furthermore, we have that IpZ, ´Ŷ q Ď Ť f RA I f p X, ´Ŷ q and Ip´Z, Ŷ q Ď Ť f RA I f p´X, Ŷ q.However, as argued above, Ť f RA I f p X, ´Ŷ q " Ť f RA I f p´X, Ŷ q " H.This concludes the proof of this last claim.Since we now know exactly how Z arises from X and Ŷ it is straightforward to check that pZ ' ´Ŷ qzA " X ' ´Y .This concludes the proof.□ Lemma 2.14.Let pE, L q any set of sign vectors and A Ď E.Then, PpL {Aq Ď PpL q{A.
Proof.Let X ' ´Y P P pL {Aq, with X, Y P L {A and I L {A pX, ´Y q " I L {A pX, ´Y q " H. Thus, that there are X, Ŷ P L with A Ď zep Xq X zep Ŷ q that otherwise coincide with X and Y , respectively.In particular, I L {A pX, ´Y q -I L p X, ´Ŷ q, since all covectors in I L p X, ´Ŷ q are 0 on A. The same holds for I L p X, ´Ŷ q and we have I L p X, ´Ŷ q " I L p X, ´Ŷ q " H.This means that X ' Y " p X ' Ŷ q{A P PpL q{A.□ We take the following from [4] and we review its proof in order to ensure that it does not rely on finiteness assumptions.Proof.We first prove the statement for deletion.To see (FS) let XzA, Y zA P L zA.Then X ˝p´Y q P L and pX ˝p´Y qqzA " XzA ˝p´Y zAq P L zA.To check (SE) let XzA, Y zA P L zA and e an element separating XzA and Y zA.Then there is Z P L with Zpeq " 0 and Zpf q " X ˝Y pf q for all f P EzSpX, Y q.Clearly, ZzA P L zA satisfies (SE) with respect to XzA, Y zA.Now, we prove the statement for contraction.Let XzA, Y zA P L {A, i.e., X XA " Y XA " ∅.Hence X ˝p´Y q X A " ∅ and therefore XzA ˝p´Y zAq P L {A, proving (FS).Towards proving (SE), let XzA, Y zA P L {A and e an element separating XzA and Y zA.Then there is Z P L with Zpeq " 0 and Zpf q " X ˝Y pf q for all f P EzSpX, Y q.In particular, if Xpf q " Y pf q " 0, then Zpf q " 0. Example 2.20.In Figure 5 the pseudolines labeled a ´1, . . ., a 3 correspond to parallel elements in the associated FAOM.
Note that a different notion of parallelism in systems of sign vectors appears in the literature, that we call here equivalence in order to avoid confusion, and that is defined by e " f if Xpeq " Xpf q for all X P L or Xpeq " ´Xpf q for all X P L .Note that this is an equivalence relation on E. Another notion is that of redundant elements, i.e., e is redundant if Xpeq " Y peq for all X P L .Definition 2.21.An AOM is called simple if all equivalence classes with respect to " are trivial and there are no redundant elements.Every AOM can be reduced to a simple one by deleting all redundant elements and all but one elements of each class of equivalent elements.The resulting AOM is a minor, that is unique up to isomorphism, called the simplification.
Remark 2.22.Let L 1 be the simplification of an AOM L .We have F pL 1 q -F pL q and F pL 1 q -F pL q.Corollary 2.23.In every simple AOM with ground set E, the reflexive closure of parallelism is an equivalence relation on E. We call πpeq the parallelism class of e P E.
Proof.Symmetry of ∥ being evident from the definition, we have to check transitivity.By way of contradiction consider three elements with e ∥ f , e ∥ g, but f and g not parallel.The restriction L rte, f, gus is a finite AOM and so L :" zepL rte, f, gusq is the geometric semilattice of flats of a (finite) semimatroid, see Lemma 3.25.Now since there is a covector X such that zepXq " tf, gu, we can find x P L with f, g P x.Now since L is simple, so is its restriction to te, f, gu.In particular, all of teu, tf u, tgu are atoms of L and x has rank at least 2 in L, so that tf u, tgu is an independent set of atoms.Now Axiom (GSL2) in Definition A.12 ensures that one among the joins teu _ tf u and teu _ tgu exists in L. But this implies existence of a covector Y P L with either te, f u Ď zepY q or te, gu Ď zepY q, contradicting our parallelism assumption.□ Remark 2.24.Our notion of parallelism could be redefined so as to allow for Corollary 2.23 to work also in non-simple AOMs, e.g., by adding to Definition 2.19 the requirement that e, f are not loops and that |te, f u X zepXq| ‰ 1 for all X P L (the latter condition implying that e, f are not parallel "in matroid sense" in the underlying semimatroid).Our choice of definition is enough since we will need it in order to study posets of covectors of AOMs, for which considering the "simple" case is no restriction of generality (see Remark 2.22).
Remark 2.25.Notice that e ∥ f if and only if, for all X, Y P L , Xpf q " Y pf q " 0 implies Xpeq " Y peq ‰ 0. This sign we can then denote by σ f peq.
Proof.The condition in the Remark's statement directly implies e ∥ f .For the other implication, suppose e ∥ f and consider X, Y P L with Xpf q " Y pf q " 0. Clearly Xpeq ‰ 0 and Y peq ‰ 0 otherwise parallelism is immediately violated.It remains to prove Xpeq " Y peq.
Indeed, if Xpeq " ´Y peq then e P SpX, Y q and we can eliminate e obtaining Z such that Zpeq " 0 and Zpf q " Xpf q ˝Y pf q " 0, a contradiction to parallelism.□ Definition 2.26.Following Huntington [21], we say that a ternary relation r¨, ¨, ¨s on a set X is a betweenness relation if it satisfies the following requirements for all a, b, c, x P X. rf, g, hs ô f, g, h are pairwise distinct and, for all X, Y P L , Xpf q " 0 and Y phq " 0 imply Xpgq " ´Y pgq, is a betweenness relation and is invariant under reorientation.
Proof.Properties (BR1) and (BR3) hold trivially.In order to prove the others let a, b, c be distinct elements of π, recall the definition of the signs σ f peq from Remark 2.25 and for x P ta, b, cu write σpxq :" ź yPta,b,cuztxu σ y pxq.
Notice that ra, b, cs is equivalent to σ a pbq " ´σc pbq, hence to σpbq " ´. (BR2) It is enough to find x P ta, b, cu with σpxq " ´.Since the AOM is simple and a, b, c are parallel, in the restriction of the AOM to the set ta, b, cu we can pick covectors X, X 1 , Y with tau " zepXq, tbu " zepX 1 q, tcu " zepY q.Now assume by way of contradiction that σpxq " `for all x P ta, b, cu.Then Xpcq " X 1 pcq, Xpbq " Y pbq, X 1 paq " Y paq and we can compute pX ' ´X1 qpaq " ´Y paq, pX ' ´X1 qpbq " Y pbq, pX ' ´X1 qpcq " 0.
Moreover, IpX, ´X1 q " Ip´X, X 1 q " H (Notice that IpX, ´X1 q " I c pX, ´X1 q, hence any W P IpX, ´X1 q must have W pcq " 0, W paq " ´Y paq and thus would witness σpaq " σ b paqσ c paq " X 1 paqp´Y paqq " X 1 paqp´X 1 paqq " ´, a contradiction.For IpX 1 , ´Xq the reasoning is analogous and contradicts σpbq " `.) Thus, by (P) the family L should contain the covector W 1 :" pX ' ´X1 q ˝Y , which satisfies W 1 paq " ´Y paq and W 1 pcq " 0 and would witness again σpaq " X 1 paqp´Y paqq " ´, a contradiction.(BR4) Let us argue again in the restriction to ta, b, cu and pick covectors X, X 1 and Y as in the proof of (BR2).If both ra, b, cs and ra, c, bs hold, then σpbq " σpcq " ´and so Xpbq " ´Y pbq and Xpcq " ´X1 pcq.In particular, by (SE) the set IpX, Y q " I b pX, Y q contains some W with W pbq " 0 and W pcq " Xpcq ˝Y pcq " Xpcq, but the latter is opposite to X 1 pcq, a contradiction to σ b pcq being well-defined.
Invariance under reorientation is apparent from the definition of r¨, ¨, ¨s.□ Corollary 2.28.Let π Ď E be a parallelism class of the given AOM.Then there is a total order ă π on π, unique up to order reversal, such that a ă π b ă π c if and only if ra, b, cs.In particular, this ordering is independent on the reorientation of the AOM.Moreover, there is a reorientation of π such that for all x, y P π, x ă π y if and only if σ y pxq " `, where σ y pxq is defined in Remark 2.25.
Proof.If π has less than 3 elements, the claim is trivial.Otherwise, recall the betweenness relation on π defined in Proposition 2.27 and choose two distinct elements e, f P π.In [21, §3.1] it is proved that the condition "a ă π b ă π c if and only if ra, b, cs" determines a pair of opposite total orderings on π, and thus letting e ă π f fully determines a total ordering of π with the desired properties.
The desired reorientation is obtained by reorienting e, f so that σ f peq " `and σ e pf q " ´, as well as reorienting every other x P π so that σ e pxq " ´if and only if e ă π x. □ Notice that the total order ă π obtained in Corollary 2.28 is unique up to order reversal.In particular, the following definition is well-posed (where we assume, after possibly reversing the order, that if an extremum exists, it is a minimum).Definition 2.29.Write 1 π resp.0 π for the unique maximal (resp.minimal) element of ă π when they exist.We will assume, after possibly reversing the order, that if an extremum exists, then a minimum exists.Corollary 2.28 allows us then to define the following partition of the ground set of an AOM, independently from the reorientation.E 01 :"te P E | both 1 πpeq and 0 πpeq existu, Example 2.30. Figure 6 illustrates different types of elements of the ground sets of FAOMS represented by affine line arrangements.

Finitary affine oriented matroids (FAOM)
We now move to a more restrictive definition, especially in order to approach a topological study of covector posets and to connect our theory with that of the unoriented "affine" version of matroids, i.e., semimatroid theory.Definition 3.1 (FAOM).A Finitary Affine Oriented Matroid is a pair pE, L q that is an AOM (i.e., it satisfies (FS), (SE), and (P)) that furthermore fulfills: . Three examples of infinite line arrangements.The FAOM associated to example (a) has E 01 " tlu, E ˚˚" th i u iPZ , E 0˚" H; the one associated to example (b) has E 01 " H, E ˚˚" th i u iPZ , E 0˚" tl j u jPN ; the one of example (c) has E 01 " E 0˚" H, E ˚˚" tl j u jPN Y th j u jPN .
Remark 3.2.Axiom (Z) implies that L is closed under infinite composition, since |zepX ˝Y q| ă |zepXq|, unless X ˝Y " X.Thus, any result of an infinite composition can be expressed as a finite composition.Moreover, (Z) also implies that |F pL q ěX | ă 8 for every X P L .
Remark 3.3.Axiom (Z) might be weakened to (Z') X P L ùñ |zepXqz Ş Y PL zepY q| ă 8.Most of the statements and proofs remain valid with some technical adjustments.We choose the stronger axiom in order to fit the "finitary" nature of the extant literature on matroids.For instance, (Z') would allow for infinitely many elements of rank 0 in the underlying semimatroid.Again, as we are interested in the structure of F pL q, omitting loops does not restrict generality.
The next lemma follows from the more general property that faces of COMs are OMs.Lemma 3.4 (See Lemma 4 in [4]).Let L be the set of covectors of an FAOM and let X P L .Then the poset F pL q ěX is isomorphic to the poset of covectors of an oriented matroid.More precisely, O :" tY |zepXq | Y ě Xu " L rzepXqs is the set of covectors of an oriented matroid on the ground set zepXq.
3.1.Topes, convex sequences and rank.Definition 3.5.The set of topes T of simple FAOM is constituted by the elements of L of full support, i.e., with X e ‰ 0 for all e P E. Remark 3.6.Note that a tope exists in a simple FAOM, since for every e P E, there is an X e P L with X e peq ‰ 0 and we can obtain a tope by composing all tX e | e P Eu by Remark 3.2.Moreover, every X P L is below some tope in in the face poset -just take any X ˝T , with T P T .Definition 3.7.The tope graph G L is the simple graph with the set T as its vertices and where a pair of vertices T , T 1 form an edge if and only if |SpT, T 1 q| " 1.In other words, G L is an induced subgraph of the hypercube Q E .
Note that (I) implies that all topes have a finite number of neighbors, since the edges of a tope T are in F pL q ďT .Together with (S), this implies the following statement.
Remark 3.8.In the tope graph G L of a simple FAOM , every vertex has finite degree and the distance of any two vertices is finite.Remark 3.9.Remark 3.8 entails in particular that the distance in G L between any two topes T, T 1 P T satisfies dpT, T 1 q " |SpT, T 1 q|.In particular, G L is an isometric subgraph of Q E , i.e., a partial cube (of finite degree and finite distances), see [4, Proposition 2].Moreover, L is uniquely determined by T and up to isomorphism by G L , see [25,Corollary 4.10].
A subset of vertices of a graph G is called convex if it contains all vertices of every shortest path between any two if its vertices.The convex hull of a subset of vertices of G is the smallest convex subgraph containing it, if it exists.A subset Remark 3.10.It is well-known, see e.g.[2], that convex subsets in partial cubes coincide with intersections of halfspaces, this is C Ď T is convex if and only if there is a sign vector X (not necessarily in L ), such that C " tT P T | T ě Xu.Moreover, for a finite subset B its convex hull convpBq can be represented by X B peq :" " 0 if e P SpT, T 1 q for some T, T 1 P B ⃝ T PB T peq otherwise.Lemma 3.11.Let L be a simple FAOM and C P T .There is an increasing sequence Proof.We consider the tope graph G L of L .Take the sequence of balls B i of radius i around C, starting with B 0 " tCu.If we define C i " convpB i q as the smallest convex subgraph of G L that contains B i , which is finite because all degrees are finite.In particular, B i is a subgraph of a finite partial cube and by Remark 3.10 C i is a subgraph of the same cube, hence finite.Moreover, by definition the subgraph induced by C i is convex and since distances are finite the sequence eventually exhausts the entire graph G L .□ Example 3.12.In Figure 7 an example of a sequence C 1 , C 2 , . . .associated to the base tope C is shaded in decreasing hue of gray.For C P T we define T pL , Cq to be the tope poset based at C, i.e, for topes T, T 1 P T we have T ď T 1 if and only if SpC, T q Ď SpC, T 1 q.Lemma 3.14.Let L be a simple FAOM, C Ď T convex and C P C .Then C is a lower ideal of the poset T pL , Cq.Moreover, if |C | ă 8, then there is a finite subset E C Ď E such that the restriction map res C : L Ñ L rE C s restricts to order isomorphisms between (1) F pL q ďC and F pL rE C sq ďres C pC q , as well as, (2) the induced subposets C Ď T pL , Cq and res C pC q Ď T pL rE C s, res C pCqq.
Proof.Let C Ď T be convex and C P C .Then by definition the set C is convex in G T .By Remark 3.10, there is a sign vector X C such that C " tT 2 P T | T 2 ě X C u. Let T ď T 1 with respect to T pL , Cq and T 1 P C .Since SpC, T q Ď SpC, T 1 q and C, T 1 ě X C also T ě X C .Thus, C is an order ideal of T pL , Cq.Now let C be finite, then since distances are finite, separators of elements of C are finite and zepX C q is finite by Remark 3.10.We define E C as the union Ť Y ďT PC zepY q.Note that this is a finite set by (I) and (Z) contains zepX C q.
The representation of C as intersection of halfspaces tT 2 P T | T 2 ě X C u yields (2).In particular, since all members of C are identical on the complement of E C , res C induces an injection from C whose restriction to res C pC q is then bijection.To see (1) note that for two covectors below C with an element e in their separator, there are also two topes T, T 1 P C with e P SpT, T 1 q and e P zepX C q. Thus, all members of F pL q ďC are identical on the complement of E C .This proves injectivity.Moreover, all elements of F pL qzF pL q ďC have an element e P E C that is in the separator with all elements of C .Thus, their restriction cannot be in F pL rE C sq ďres C pC q .Since res C from F pL q to F pL rE C sq is surjective by definition, this proves surjectivity.□ Corollary 3.15.The poset F pL q _ (obtained from F pL q by adding a global minimum and a global maximum, cf.Appendix A.1.1) is graded of finite length.
Proof.We have to prove that any two maximal chains in F pL q _ have the same, finite length.Let ω, ω 1 be two such chains and write X :" maxpωzt p 1uq, X 1 :" maxpω 1 zt p 1uq.Let i be such that X, X 1 P C i as in Lemma 3.11.Then by Lemma 3.14 both ω and ω 1 are maximal chains in the poset F pL rE C i sq _ that is graded of finite length by [8,Theorem 4.5.3].□ Remark 3.16.From Corollary 3.15 follows immediately that F pL q is ranked of finite length.
Corollary 3.17.The poset F pL q is the poset of cells of a regular CW-complex that we call KpL q.The dimension of KpL q is the length of F pL q.
Proof.By [8, Proposition 4.7.23], it is enough to check that, for every X P F pL q, the interval I X :" F pL q ăX is homeomorphic to a rkpXq-sphere.Now by Lemma 3.14 I X is an order ideal in F pL rE C sq, for some finite, convex set of topes C (whose existence is ensured by Lemma 3.11).In particular, I X is an order ideal in the "bounded complex" of the finite affine oriented matroid F pL rE C sq (see [8,Definition 4.5.1]).With [8, Discussion before 4.5.7,§4.3] the claim follows.□ 3.2.Shellings.As a stepping stone towards determining the topology of covector posets, we prove their shellability.A brief introduction to shellability as well as to the parts of the theory we need here are sketched in the Appendix ( §A.1.2,§A.1.3),where we also point to some literature for further background.
Proposition 3.18.Let L be the set of covectors of an FAOM.Then, the poset F pL q _ admits a recursive coatom ordering without critical chains.In particular, both the regular CW-complex KpL q and its barycentric subdivision, the simplicial complex ∆pF pL qq, are shellable and contractible.
Proof.Let C be any tope of L and choose a sequence C 1 Ĺ C 2 Ĺ . . .according to Lemma 3.11.Since for all i the set C i is a lower ideal of T pL , Cq, it is possible to choose a linear extension ă of T pL , Cq such that T i ă T j if T i P C i , T j P C j and i ă j.Then ă is a total order, and it is a well-ordering because for any X Ď T pL , Cq the set ti | C i X X ‰ Hu Ď N has a minimum, say i 0 , and the set C i 0 X X is finite, hence has a ă-minimum T , that is also the minimum of X .We show that ă defines a recursive coatom ordering on the poset F pL q _.The set of coatoms of F _pL q is exactly the set of topes.So let T be a tope, and let i be such that T P C i .For T 1 , T2 P res C i pC i q let T 1 ă i T 2 if and only if res ´1 C i pT 1 q ă res ´1 C i pT 2 q.Since res C i pC i q is a lower ideal in T pL rE C i s, res C i pCqq, the order ă i can be extended to a linear extension of T pL rE C i s, Cq where the elements of res C i pC i q come first.By [8, Proposition 4.5.6],this defines a recursive coatom ordering ă i of F _pL rE C i sq.For every tope T 1 of L rE C i s we let Q i T 1 be the associated distinguished set of coatoms of F pL rE C i sq ďT 1 , uniquely determined by ă i .Now, for any T P T pL , Cq we can set Q T :" res ´1 C i Q i res C i pT q where i is such that Thus, by Remark A.5, }F pL q} is shellable.Now if some chain ω Ď F pL q is critical, then it is critical also in the shelling of F pL rE C i sq, where i is such that max ω P C i .But we know [8, Theorem 4.5.7] that F pL rE C i sq is contractible, hence no shelling of the latter poset has critical chains.Therefore the obtained shelling of }F pL q} has no critical cells either, and this complex is contractible as well.The claim about KpL q follows with [8, Lemma 4.7.18].□ 3.3.Topology of covector posets.We study the topology of the order complexes of posets of covectors of FAOMs.For basic terminology and notations about combinatorial topology we refer again to the Appendix A.1.
Lemma 3.19.Let L be the set of covectors of an FAOM.Let ω be a maximal chain in F pL q, let X P ω and write ω 1 :" ωztXu.Let Y be the set of all Y P L such that ω 1 Y tY u is a chain in F pL q.Then and recall that the poset of covectors of a finite AOM is a filter inside that of a finite oriented matroid).Now we prove (2).Consider any chain γ in F .Recall that shellability of F pL q implies shellability of links, that F pL q being ranked implies that the link of γ is pure of dimension k :" d ´dim }γ}, and that (1) implies that every ridge of Lkpγq is contained is at most two facets of the link (see §A.1.2).Moreover, by (Z) and (I) the complex Lkpγq is finite.Now [8, Proposition 4.7.22]applies, implying that Lkpγq is either a sphere or a closed ball, the second case entering exactly if there is one ridge that is contained in only one facet.Equivalently (e.g., by [31, p. 7]) }γ} is in the boundary of }F pL q} if and only if γ Ď ωztXu for a maximal chain ω Ď F pL q and X P ω such that |Y | "  1 then }F pL q} is homeomorphic to a 2-disk.On the other hand, if L is the FAOM of the arrangement in Figure 6.(b) then }F pL q} is a tiling of the plane by squares, i.e., it is homeomorphic to R 2 .Figure 6.(a) shows an example outside the scope of Theorem 3.20: there, }F pL q} is the poset of cells of an infinite row of juxtaposed squares.See Figure 8. 3.4.The underlying semimatroid.We show that zero sets of covectors of an FAOM define a semimatroid on the same ground set.We point to the Appendix A.2 for some basic notions and notations of semimatroid theory that we will be using in this section.
Definition 3.22.Given a system of sign vectors pE, L q define LpL q :" tzepXq | X P L u KpL q :" ď APLpL q 2 A .
Elements of LpL q are called flats, elements of KpL q "central sets" of L .For A, B P LpL q, let A ď B :ô A Ď B and define F pL q :" pLpL q, ďq.
Remark 3.23.Notice that X ď Y implies zepXq Ě zepY q, where ď is the partial order of F .Therefore, taking zero sets induces an order reversing poset map zep¨q : F pL q Ñ F pL q.
Example 3.24.Figure 9 depicts the poset F pL q and LpL q where L is the set of covectors of the line arrangement in Figure 1.
Figure 9.The poset F pL q (l.-h.s.) and the poset LpL q (r.-h.s.) for the FAOM L corresponding to the arrangement of Figure 1.
Lemma 3.25.Let pE, L q be a finite AOM.Then F pL q Ď 2 E is the geometric semilattice of flats of a finitary semimatroid S pL q.This underlying semimatroid is unique up to isomorphism.
Proof.By Remark 2.3, pE, L q is a finite affine oriented matroid in the sense of [8].In this case, L can be embedded into the set L peq of covectors of an oriented matroid on the set E Y teu where e is a new element that is not a loop, so that there is a unique atom e of F pL peq q containing teu.Moreover, e.g. by [8, Section 10.1], F pL q " F pL peq qzF pL peq q ěe .
Since F pL peq q is a geometric lattice [8, Proposition 4.1.13],by [33, Theorem 3.2] F pL q is a geometric semilattice.□ Lemma 3.26.Let L be the set of covectors of an AOM on the ground set E.
(1) For every F 1 , F 2 P LpL q, F 1 X F 2 P LpL q.
(2) For every A Ď E, LpL rAsq " tG X A | G P LpL qu.
(3) For all A P LpL q, F pL rAsq " F pL q ďA .( 4) For all A Ď E, the assignment X Þ Ñ ιpXq :" min ď tG P F pL q | G X A " Xu defines an order preserving embedding ι : F pL rAsq ãÑ F pL q. (5) For all A Ď E and all F 1 , F 2 P F pL rAsq, if the join F 1 _ F 2 exists in F pL rAsq, then ιpF 1 q _ ιpF 2 q exists in F pL q.
Proof.In order to check (1) let X 1 , X 2 P L such that F i " zepX i q for i " 1, 2. Then axiom (C) ensures X 1 ˝X2 P L .Thus F 1 X F 2 " zepX 1 ˝X2 q P LpL q.Now, by definition, F P LpL rAsq if and only if there is X P L with F " zepX |A q " zepXq X A, i.e., if and only if F " G X A for some G P LpL q (indeed, G P LpL q if and only if there is X P L with zepXq " G).This proves (2).Item (3) follows from (2) since, by (1), F pL q ďA " tG X A | G P LpL qu.
The assignment in ( 4) is well-defined by ( 1) and clearly determines an order preserving function.Its injectivity follows from the existence of the right-sided inverse G Þ Ñ G X A (this is well-defined by ( 2)).
For item (5) let F 1 , F 2 have a join F 1 _ F 2 in F pL rAsq.Then ιpF 1 _ F 2 q ě ιpF i q for i " 1, 2, hence the set of upper bounds of ιpF 1 q and ιpF 2 q is nonempty and, by (1), has a unique minimal element.□ Theorem-Definition 3.27.Let pE, L q be an AOM satisfying (Z).Then F pL q Ď 2 E is the geometric semilattice of flats of a finitary semimatroid S pL q.This underlying semimatroid is unique up to isomorphism.
Proof.By definition F pL q is partially ordered by inclusion.By (Z) F pL q is a chain-finite subposet of P fin pEq.By Theorem A.13, it is enough to prove that F pL q is a geometric semilattice whose order ideals are lattices of flats of matroids.If |E| ă 8, the claim is proved as Lemma 3.25.The proof for general E follows from the fact that the obstructions to the claim can be detected in a finite restriction.More precisely, first notice that F pL q is bounded below, with p 0 F pL q " te P E | @X P L | Xpeq " 0u, which is well-defined because topes exist (by Remark 3.6).Now checking that F pL q is graded and that order ideals are lattices of flats of matroids (in particular, then it satisfies (GSL1)) amounts to checking statements about order ideals of the type F pL q ďU which, by Lemma 3.26.(3),are isomorphic to order ideals of (finite) geometric semilattices F pL rU sq, where the claims hold.
The poset F pL q is a meet-semilattice by Lemma 3.26.(1).In order to check (GSL2) let A 1 , . . ., A k be an independent set of atoms in F pL q that joins to some U " _A i P F pL q and let Y P F pL q have rank less than U .Notice that A 1 , . . ., A k is an independent set of atoms in F pL rU Y Y sq: indeed for all I Ď rks the inclusion _ iPI A i Ď U implies _ iPI A i " ιp_ iPI A i q P F pL rU YY sq by Lemma 3.26.(2,4).Thus, by Lemma 3.25 in the finite restriction F pL rU YY sq there is some i such that A i _Y exists in F pL rU YY sq, and so by Lemma 3.26.(5) this join also exists in F pL q. □ Corollary-Definition 3.28.For every AOM pE, L q satisfying (Z), the set KpL q from Definition 3.22 is the set of central sets of the underlying semimatroid, i.e., KpL q " KpS pL qq.
Proof.Immediate from Theorem A. 13. □ Corollary 3.29.Let pE, L q be an AOM satisfying (Z) and let B be any basis of the semimatroid S pL q.For every b P B choose b 1 P πpbq.Then B 1 :" tb 1 | b P Bu is a basis of S pL q.
Proof.Induction on the rank of S pL q.If the rank is 0 or 1, there is nothing to prove.Let then S pL q have rank r ą 1 and let B, B 1 be as in the claim.Choose b 0 P B. By induction hypothesis applied to L r Ť b‰b 0 πpbqs, B 1 zb 1 0 is an independent set of S pL q of rank r ´1.Now apply (CR2) to the sets B and B 1 zb 1 0 .Since every element of B except b 0 is parallel to some element of B 1 zb 1 0 , the only way for (CR2) to hold is that B 1 is central in S pL q (cf.§A.2 for terminology) and has rank r. □ Corollary 3.30.Let pE, L q be an AOM satisfying (Z) and let B be any basis of the semimatroid S pL q.Then there is a unique X B P max F pL q with B Ď zepX B q.
Example 3.31.In Figure 5 the elements a 0 and b 0 form a basis B " ta 0 , b 0 u of the FAOM represented by the given pseudoline arrangement.The covector X B corresponds to the vertex highlighted in white.
Proof of Corollary 3.30.Since B is a basis, clpBq exists and is a maximal element in F pL q.
In particular, there is X B P max F pL q with B Ď zepX B q.In order to prove uniqueness consider any Y P L with B Ď zepY q.If Y ‰ X, there is some e P E with Y peq " ´Xpeq ‰ 0.
Elimination of e from X and Y would give a Z ŋ X in F pL q, contradicting maximality of X. □ 3.5.Rank.We briefly compare the different notions of rank that have appeared so far and set some notation for the remainder of the paper.
Proposition 3.32.The order reversing map zep¨q : F pL q Ñ F pL q from Remark 3.23 is rank-preserving.
Proof.In analogy with the proof of [8, Proposition 4.1.13.(ii)], it is enough to consider two X, Y P L with X Ì Y in F pL q and to prove zepXq Í zepY q in F pL q.Now, for such X, Y clearly zepXq Ľ zepY q.If there is a covector Z P L with zepXq Ľ zepZq Ľ zepY q, then we can in fact choose this covector to be in the upper interval F pL q ěX (take for instance X ˝Z), contradicting [8, Proposition 4.1.13.(ii)] for the OM L rzepXqs (cf.Lemma 3.4).□ Definition 3.33.For any given FAOM L , we will henceforth write rk for both the rank function of F pL q and for the rank function of its underlying semimatroid.We write rkpL q " rkpS pL qq for the rank of either (i.e., the length of F pL q and F pL q).Corollary 3.34.Let L be an FAOM and let A Ď E.Then, rkpL rAsq " rkpAq and rkpL {Aq " rkpL q ´rkpAq.
Proof.These identities are known on the semimatroid level, see, e.g., [17, §1.1] □ 3.6.Order type of parallelism classes.In this section we show that the isomorphism type of the natural total orderings of parallelism classes of an FAOM (described in Corollary 2.28) is restricted.Recall the notation and conventions of Definition 2.29.
Lemma 3.35.Suppose L is the set of covectors of an AOM satisfying (S) and let e P E. Then pπpeq, ď πpeq q is order isomorphic to Z if e P E ˚˚, to N if e P E 0˚, and to the segment t0, 1, . . ., n |πpeq| u Ď N if e P E 01 .
Example 3.36.Recall the FAOMs of Figure 6.In case (a) we have that |πplq| " 1.In both cases ď πph 0 q has the order type of Z.In case (b) the order type of ď πpl 0 q is N Proof of Lemma 3.35.Write π for πpeq and ă for ă πpeq .Recall that by Corollary 2.28 we have that π is a total order.Given x ă y in π choose covectors X and Y with x P zepXq and y P zepY q.Now any z P π with x ă z ă y must have Xpzq " ´Y pzq ‰ 0 and thus z P SpX, Y q.By (S) there are at most finitely many such z's.Now, if e P E 01 the claim follows by taking x " p 0 π , y " p 1 π .Otherwise, for every x P π there is some y P π with y ą x and so the assignment spxq :" mintz P π | x ă z ď yu determines a well-defined "successor" function s : π Ñ π (the set on the r.h.-s. is finite by the previous discussion, nonempty since it contains at least y, and independent on the choice of y since spxq is an immediate successor of x and ď is a total order).Analogously, for every y P π, y ‰ p 0 π , we can find an element s ´1pyq :" maxtz P π | p 0 π ď z ă yu so that s ´1pspxqq " x for all x P π and thus s is injective.
Claim.For every x P π, the function f : N Ñ π ěx , n Þ Ñ s n pxq is bijective.Proof.Injectivity of s implies injectivity of f .In order to prove surjectivity, let y P π ěx .As above, by (S) there is a finite number, say k, of z P π, x ă z ď y, therefore s k pxq " y.
If e P E 0˚, in the claim above we can choose x " p 0 π and we obtain an order isomorphism between pπ, ď π q and N with the natural order.If e P E ˚˚, the Claim above gives an order isomorphism between N and π ěe and an order antiisomorphism between N and π ďe , that combine to an order isomorphism between pπ, ď π q and Z with the standard ordering, where e is mapped to 0. □ Corollary 3.37.Suppose that L is an AOM satisfying (S) and let X P L .Then there is a reorientation of L such that for every parallelism class π, there is a unique element δ X pπq P π such that, for every e P π, e ă π δ X pπq ñ Xpeq " `, e ą π δ X pπq ñ Xpeq " ´, Xpδ X pπqq P t0, ´u Proof.Consider the reorientation of L given in Corollary 2.28.First note that if Xpπq Ď t0, ´u then δ X pπq :" 0π will do.Otherwise, we prove that the maximum m π pXq :" max ďπ te P π | Xpeq " `u exists.For this, choose an f P π and any Y P L with f P zepY q.Then by Corollary 2.28 we know that Y peq " `if e ă π f and Y peq " ´if e ą π f .Now assume by way of contradiction that the stated maximum m π pXq does not exist.Then, either Xpeq " ´for all e P π (a case that we excluded at the beginning), or there are infinitely many e ą π f with Xpeq " `, violating (S) between X and Y .Now, the element δ X pπq :" spm π pXqq, the successor of δ X pπq, satisfies the claim.□ Example 3.38.In Figure 7 with the pictured reorientation and choosing the ordering l i ă πpl 0 q l j , resp h i ă πph 0 q h j if and only if i ă j, for the covector X corresponding to the bold segment we have δ X pπph 0 qq " h 1 and δ X pπpl 0 qq " l 1 .For the tope C we have δ C pπph 0 qq " h 1 and δ C pπpl 0 qq " l 0 .

Basis frames and embeddings into Euclidean space
This section contains some fundamentals that will be needed at a later stage, and can be skipped in a first reading.The goal is to study the homeomorphism type of covector posets of FAOMs by comparing them with those of restrictions to unions of parallelism classes of elements of a basis.

4.1.
Covectors of restrictions to basis frames.Throughout this section suppose that L is the set of covectors of an AOM on the ground set E satisfying (S) and (Z).In particular, (Z) ensures that there is a well-defined underlying semimatroid S pL q (see Corollary 3.28), while (S) implies that the canonical ordering of parallelism classes has the order type of subsets of Z (see Lemma 3.35).Throughout this section let B be a basis of the semimatroid S pL q.We start by describing an explicit model of the restriction L r r Bs.Without loss of generality, suppose that pE, L q is reoriented so to satisfy Corollary 2.28.For every parallelism class π of L we fix an order isomorphism (1) where n π :" |π| and an index set (2) Moreover, our chosen reorientation is the one that yields Corollary 3.37 and so for every X P t`, ´, 0u r B we can define a vector i X P 1 2 Z πpBq as (5) i X pπq :" 3. With the definitions above, the following hold.
(1) For every X P t`, ´, 0u r B , X i X " X.
(2) For every i P 1 2 Z πpBq , i X i " i.
Proof.We check both identities elementwise.
(1) Fix e P r B. By definition, X i X peq " `if and only if i X pπpeqq ą j πpeq peq, i.e., j π pδ X pπqq ą j πpeq peq or, equivalently, δ X pπq ą e, and by Corollary 3.37 the latter means Xpeq " `.The cases X i X peq " 0 and X I X peq " ´are analogous.
(2) Notice first that j π pδ X i pπqq " ripπqs for all i, with j π pδ X i pπqq " ipπq if and only if X i pδ X i pπqq " 0. Now we compute with the definition, for every parallelism class π: Define a partial order ĺ on 1 2 Z by setting p ă q if and only if p P Z, q " ˘1 2 .(See Figure 10.)This restricts to a partial order on I π , for every parallelism class π, and induces a partial order on IpBq by taking Cartesian product of the orderings of all I π .This ordering on IpBq we also call ĺ, and it can be explicitly described as i ĺ i 1 if and only if i πpbq ĺ i 1 πpbq for all b P B.
Proof.We first prove that the assignment is well-defined.By Corollary 3.30, if every coordinate of i is an integer then X i P L r r Bs.Otherwise, there is an i ˚ĺ i with all integer coordinates, and we have X i ˚P L r r Bs, with X i ˚peq " X i peq if e R zepX i ˚q.Now B ˚:" zepX i ˚q is a basis of L r r Bs and thus L rB ˚s is a Boolean (finite) oriented matroid, i.e., L rB ˚s " t`, ´, 0u B ˚and, in particular, there is Bs as desired.This proves that the function is well-defined.The stated inverse is clearly well-defined, and Lemma 4.3.(2) shows that it is, in fact, an inverse.□ We use this bijection in order to decompose the covector poset into a Cartesian product.
Theorem 4.6.The function i Þ Ñ X i defines an isomorphism of posets pIpBq, ĺq -F pL r r Bsq.
Proof.We prove that both i Þ Ñ X i and X Þ Ñ i X are order preserving.First suppose i ĺ i 1 in IpBq.We prove X i ď X i 1 componentwise.Consider a parallelism class π.If i π " i 1 π , then obviously X i peq " X i 1 peq for all e P π.Otherwise, (1) i π P Z and (2) i 1 π " i π ˘1 2 .This means that, with e :" j ´1 π pi π q, we have (1) X i peq " 0 and (2) X i 1 peq ‰ 0, with X i 1 pf q " X i pf q for all f P πzteu.Therefore pX i q |π ď pX i 1 q |π , and repeating the argument for every π proves X i ď X i 1 .
Now suppose X ď Y in F pL q, i.e., Xpeq ď Y peq for all e P E. In particular, Xpeq " Y peq whenever Xpeq ‰ 0. We prove the inequality i X ĺ i Y componentwise.Fix a parallelism class π.If Xpeq ‰ 0 for all e P π, then i X pπq " i Y pπq.Otherwise there is e P π with Xpeq " 0 (which means i X pπq " j π peq), and by definition of parallelism, Xpf q ‰ 0 (hence Y pf q " Xpf q) for all f P πzteu.Y peq can now be 0, `or ´and, depending on those cases, we will have i Y pπq " j π peq, i Y pπq " j π peq `1 2 , i Y pπq " j π peq ´1 2 .In any case, i X pπq ĺ i X pπq.□ Example 4.7.If L is the FAOM of the line arrangement of Figure 2 and B " ta 0 , b 0 u is, e.g., the basis described in Example 3.31, then Theorem 4.6 states the isomorphism between the FAOM F pL p r Bqq (represented by the bold-faced lines in Figure 5) and pIpBq, ĺq, which is the FAOM of the arrangement in Figure 6.(b).

4.2.
Embeddings into Euclidean space.Throughout this section let L be the covector set of a simple FAOM of rank d and let B be a basis of the semimatroid S pL q such that B Ď E ˚˚.For every π P πpBq we fix an arbitrary order isomorphism j π : π Ñ Z and, thus, an isomorphism F pL r r Bsq Ñ IpBq.
Notation 4.8.For brevity, in this section we write F for F pL q and F r B for F pL r r Bsq.
Proof.The poset-isomorphism from Theorem 4.6 establishes a one-to-one correspondence between chains and, thus, an isomorphism of simplicial complexes.□ Proof.The elements of IpBq are the barycenters of the cells of the cube complex Q of integer unit cubes, with set of vertices Z d Ď R d , and i 1 ĺ i 2 in IpBq if and only if the cell with barycenter i 1 is contained in the cell with barycenter i 2 .Therefore }IpBq} is realized as a geometric simplicial complex by the barycentric subdivision of Q, see Remark A.6.In particular, the homeomorphism in the claim can be defined by affine extension of the inclusion of IpBq into R d .□ Lemma 4.11.Let L be the covector set of a simple FAOM and let B be a basis of the semimatroid S pL q such that B Ď E ˚˚.Consider the natural restriction map which induces an order preserving, surjective map F Ñ F r B .For every Y P F r B , the following hold.
(2) The poset F ďϕ ´1pY q has length equal to rkpY q.
(3) The order complex }F ďϕ ´1pY q } is a subdivision of a rkpY q-dimensional PL-ball with boundary }ϕ ´1ppF r B q ăY q}.
Proof.Let L , B and Y be as in the claim.Moreover, we assume that L is reoriented as to satisfy Corollary 3.37.As all three items are claims regarding preimages of pF r B q ďY , by passing to contractions and simplifying, it is enough to consider the case Y P max F r B , i.e., Y is a tope of L r r Bs.Then zepY q " H because L is simple.Thus if k P Z πpBq is defined by k π :" j π pδ Y pπqq ´1 for all π P πpBq, Corollary 3.37 yields, for all e P E, Y peq " `if j πpeq peq ď k π , Y peq " ´if j πpeq peq ą k π .
In particular, since B Ď E ˚˚, the minimal elements of pF r B q ďY are given as where ϵ ranges in t0, 1u πpBq and Y " ⃝ ϵPt0,1u πpBq X k`ϵ (the composition taken in any order), as can be explicitly verified e.g., via the isomorphism of Theorem 4.6.Now, for every ϵ P t0, 1u πpBq let B ϵ :" zepX k`ϵ q.Every B ϵ is a basis of the semimatroid S pL r r Bsq, and hence also of S pL q, because r B has maximal rank.Thus by Corollary 3.30 for every ϵ there is a unique cocircuit Z ϵ P min F pL q with ϕpZ ϵ q " X k`ϵ .See the top part of Figure 11 for an illustration of the setup.Let B ˝" Ť πPπpBq j ´1 π ptk π , k π `1uq.It is apparent that L rB ˝s is the set of covectors of the arrangement given by the hyperplanes x π " k π and x π " k π `1 in R πpBq ("the facet-defining planes of a cube").In this interpretation, the restriction Y 1 :" Y |B ˝is the only bounded tope of L rB ˝s (cf.Definition A.22).Now by Lemma A.23 we have that anything that maps to Y 1 (in particular, Y itself as well as every covector in L ďϕ ´1pY q ) is bounded in L .
Ad (1).We argue by induction on the rank of L .If the rank is 0, then B " H and the claim is trivial.Let then L have rank r ą 0 and consider the following statement.p:q For every W P L with ϕpW q " Y and every e P zepW q we have that e P SpZ ϵ 1 , Z ϵ 2 q for some ϵ 1 , ϵ 2 P t0, 1u πpBq .
Let us first check that proving p:q is enough.First notice that the claim, together with the fact that there are finitely many Z ϵ , ensures immediately that there can be only finitely many e P E such that there is W P L with ϕpW q " Y and zepW q " teu, since otherwise axiom (S) would be contradicted for some pair of cocircuits of the form Z ϵ .Now, for every such e P E, the induction hypothesis applied to the contraction L r r B Yteus{e shows that there are finitely many W with ϕpW q " Y and zepW q " teu.This ensures that ϕ ´1pY q has finitely many elements of rank pr ´1q.
By the second statement in Remark 3.2, it follows that there can be at most finitely many topes in ϕ ´1pY q.Moreover, (I) implies that there can be at most finitely many maximal elements below any given W P L .Thus ϕ ´1pY q has finitely many elements.
Proof of p:q.
Let W P L be such that ϕpW q " Y , let e P zepW q.Then the image W 1 of W in L rB ˝Y teus is not maximal and we can choose T 1 P L rB ˝Y teus maximal with T 1 ě W 1 .Let then T 2 :" W 1 ˝´T 1 , which is a maximal element of L rB ˝Y teus ěW 1 by (FS).See the lower part of Figure 11 for illustration.Then all of W, T 1 , T 2 map to Y 1 , and are therefore bounded.In particular, by [8, Proposition 3.7.2]we have that T 1 and T 2 can be obtained from W 1 by composing with some cocircuits of L rB ˝Y teus.Now the only cocircuits with nontrivial composition with W 1 are those that are nonzero on e, hence exactly the images Z 1 ϵ of the Z ϵ 's, for ϵ P t0, 1u πpBq .Since T 2 peq " ´T 1 peq ‰ 0, there must then be ϵ 1 and ϵ 2 with Z 1 ϵ 1 peq " ´Z1 ϵ 2 peq ‰ 0, and hence Z ϵ 1 peq " ´Zϵ 2 peq ‰ 0 as well.
Since by Axiom (I) intervals in L r r Bs are finite, part (1) implies that L ďϕ ´1pY q is finite as well.Write for brevity P :" F pL q ďϕ ´1pY q .This is a convex set, as it coincides with all X P L such that p;q

#
Xpeq P t0, `u if e P r B and j πpeq peq " k πpeq , Xpeq P t0, ´u if e P r B and j πpeq peq " k πpeq `1.
In particular, with Lemma 3.14 we can find a finite E 1 Ď E such that P is an order ideal in in the poset of covectors of the finite affine oriented matroid L rE 1 s.We can then extend L rE 1 s to the set O of vectors of a (finite) oriented matroid with one additional element g (see Remark A.21).In O, P is a convex set defined by the above equations, in addition to Xpgq " `.Such a convex set of covectors, with the usual partial order, is known to be a shellable poset [8,Proposition 4.3.6].Moreover, since P it is not the full poset of covectors of a contraction of O (e.g., because P Ě ϕ ´1pY q and the latter contains at least one tope, since ϕ is order preserving), again by [8, Proposition 4.3.6]we conclude that p::q the order complex of P is is a shellable PL-ball of dimension r.
Ad (2).The length of a poset equals the dimension of its order complex, hence (2) follows directly from p::q.
we can define a sign vector g.X by setting g.Xpeq :" Xpg ´1peqq for all e P E. This extends the action of G on E to an action on the set of sign vectors.For X Ď t`, ´, 0u E write g.X :" tg.X | X P X u.
Definition 5.1.A group G acts on an AOM pE, L q if G acts by permutations on E and g.L " L for all g P G.An action of G on L will be denoted by α : G oe L .The action of G is called sliding if gpeq P πpeq for all e P E.
Example 5.2.The notion of sliding actions is meant to model the action of a group of translations on an arrangement.For instance consider the pseudoline arrangement of Figure 5.The action that corresponds to a vertical "translation" of the picture sending a i to a i`1 is sliding.On the other hand, the reflection with respect to a vertical line passing through, e.g., the point where b 0 meets a 3 is not sliding, since it switches pairs of diagonal pseudolines passing through that point, and so any of those will not be parallel to its image.
Lemma 5.3.Let pE, L q be an AOM and let τ P t`, ´uE .Every G-action on pE, L q induces a G-action on the reorientation L pτ q .If the action on L is sliding, so is the induced action on L pτ q .
Proof.Given any group action of G on L and any h P t`, ´uE , G acts on the reorientation L pτ q by g.pτ ¨Xq " τ ¨g.X, for all g P G (recall Definition 2.8).The claim about sliding actions follows because parallelism is preserved under reorientation, as zepXq " zepτ ¨Xq for all X P L .□ Lemma 5.4.Let pE, L q be an AOM with an action α of a group G. Let A Ď E.
(2) If GA=A, then α restricts to an action α rAs : G oe L rAs.
If α is sliding, then so are α {A and α rAs .
Proof.The check of the claims is straightforward.□ Lemma 5.5.Every action of a group G on an FAOM pE, L q induces an action of G on the underlying semimatroid S pL q.If the action is sliding, the action on S pL q is translative.
Proof.Let e P E and g P G.Then, the sliding property implies e ∥ gpeq.In particular, with Corollary 3.28 we have te, gpequ R KpL q whenever e ‰ gpeq, and the claim follows.□ Every action α : G oe L induces an action α : G oe F pL q by poset automorphisms.We will now begin the study of quotients of this poset.Recall that every poset can be seen as an acyclic category (see Appendix A.1.4).Definition 5.6.Let α : G oe L be a group action on an AOM.Let q α : F pL q Ñ F pL q G be the quotient functor in the category AC.Explicitly, if we regard G as a one-object category G with one morphism for each group element and the group operation as composition law, then α defines a diagram in AC over G, and q α is the universal co-cone over this diagram with target the diagram's colimit F pL q G. Remark 5.7.Since, by Corollary 3.15, F pL q is ranked, the category F pL q G can be described explicitly via [16,Lemma A.18].It has object set ObpF pL q Gq " ObpF pL qq{G, the set of orbits of objects.The morphisms of F pL q G are orbits of morphisms of F pL q: the orbit of ϕ : X Ñ Y is a morphism Gϕ : GX Ñ GY and composition between orbits Gϕ and Gψ is defined as the orbit of the composition of ϕ and ψ, when it exists.
Lemma 5.8.Let α : G oe L be a group action on an AOM.The following hold for every A P KpL q and g P G.
Proof.The first statement can be checked straightforwardly.The second follows from Lemma 5.5, as any X P L {A X L {gpAq would have zepXq Ě A Y gpAq, but translativity of the induced action on S pL q implies that A Y gpAq can only be a central set if gpAq " A.
For the third statement, we regard F pL q as an acyclic category and we denote by X ď Y the unique morphism between any two X, Y P F pL {Aq, if one exists.Now the assignment (6) Morpq α pF pL {Aqqq Ñ Morpq α {A pF pL {Aqqq, GpX ď Y q Þ Ñ stabpAqpX ď Y q is well-defined since any representative pX 1 ď Y 1 q P GpX ď Y q has the form pX 1 ď Y 1 q " gpX ď Y q for some g P G, and if g R stabpAq then g.X, g.Y R F pL {Aq by part (2).The proof of item (3) is complete by noticing that (6) describes the inverse of the natural functor q α {A pF pL {Aqq Ñ q α pF pL {Aqq induced by the inclusion stabpAq Ď G.

Actions on parallelism classes.
Lemma 5.9.If the action of G on pE, L q is sliding, then the restriction of the action on each parallelism class π is an action by order isomorphisms of the ordered set pπ, ă π q.
Proof.Let π be a parallelism class of pE, L q. Since the definition of ă π is independent on reorientation, by passing to a suitable reorientation of L (as in Corollary 2.28) and recalling Remark 2.25 we can assume that e ă π f if and only if σ f peq " `, for all e, f P π.Now let g P G and consider any two X, Y with Xpgpf qq " Y pgpf qq " 0. This means g ´1.X pf q " g ´1.Y pf q " 0 and, since g ´1.X, g ´1.Y P g ´1.L " L , the order relation e ă π f implies g ´1.X peq " g ´1.Y peq " σ f peq " `.Since Xpgpeqq " g ´1.X peq and Y pgpeqq " g ´1.Y peq, we conclude σ gpf q pgpeqq " Xpgpeqq " Y pgpeqq " `and so gpeq ă π gpf q.
Theorem 5.16.Let pE, L q be a nonempty FAOM with a distinguished basis B Ď E ˚,˚, and let a free abelian group G -Z |B| act on pE, L q so that the induced action on L r r Bs is free.If the action is sliding, then }F pL q G} is homeomorphic to the |B|-torus pS 1 q |B| .Proof.First note that, by Remark 5.15, there is a homeomorphism }F pL q{G} » }F pL q}{G.Moreover, the restriction map L Ñ L r r Bs is G-equivariant, and so is the homeomorphism of nerves that it induces (see Proposition 4.12).Therefore the quotients }F pL q}{G and }F r B }{G are homeomorphic.Analogously we have a homeomorphism with }IpBq}{G, where the action is the one described in Corollary 5.12.Now Lemma 4.10 gives a concrete realization of }IpBq} as R |B| , and on this realization the G-action described in Corollary 5.12 corresponds to the action of the subgroup kpGq of the group of integer translations Z |B| .Since stabilizers of kpGq are trivial (by the freeness assumption), the subgroup kpGq is full-rank, and so R |B| {kpGq is homeomorphic to a |B|torus, as was to be proved.□ Corollary 5.17.Let α : G oe L , let B be as in the statement of Theorem 5.16, and let A P KpL q.Then, Q α p}F pL {Aq}q is homeomorphic to a p|B| ´rkpAqq-dimensional torus.
Proof.We check that the action α {A satisfies the hypotheses of Theorem 5.16; the claim about the dimension follows with Corollary 3.34.The action is sliding by Lemma 5.4 and free, e.g., by Remark 2.11.We claim that there is a basis B 1 of L {A with B 1 Ď B. In fact, by (CR2) in the semimatroid S pL q any maximal independent subset I Ď A can be completed to a basis J of S pL q using elements from B, and it is enough to take B 1 :" JzI.Now for every b 1 P B 1 and every b 2 P πpb 1 q we have, again by (CR2), that A Y tb 1 u P KpL q.Moreover, b 1 R clpAq implies A Y tb 2 u P KpL q and b 2 R clpAq.Therefore b 2 ∥ b 1 in L {A, and so the parallelism class of b 1 in L {A contains the parallelism class of b 1 in L .In particular, B 1 P pEzAq ˚,˚.We are left to check that stabpAq acts freely on pL {Aqr Ă B 1 s which is straightforward.□ 5.3.Toric pseudoarrangements.Throughout this section let α : G oe L denote a free and sliding action of a finitely generated free abelian group G on an FAOM L , and suppose that S pL q has a basis B P E ˚,˚.
Let T :" }F pL q}{G and, for every a P E{G, let where e is any representative of a (this is well-defined by Lemma 5.8.( 1)).We then let A α :" tT a | a P E{Gu.
We call any such arrangement of subspaces of the torus T a toric pseudoarrangement.If S pL q has no loops, A α is called proper Example 5.19. Figure 3 shows the toric pseudoarrangement obtained by quotienting the pseudoarrangement of Figure 2 by the action of Z 2 by translations whose fundamental region is shaded in Figure 2.
Lemma 5.20.Every toric pseudoarrangement A α as in Definition 5.18 defines a CWcomplex structure KpA α q on T , with one cell for every object in q α pF pL qq.
If A α is proper, the union YA α is the pd ´1q-skeleton of KpA q, here d " dim T " rkpL q.In particular, the complement of YA α in T is a union of open d-cells.
Proof.For every GX P Obpq α pF pL qqq, by [16,Lemma A.19] freeness of the action implies that the slice category below GX (see [16,Definition A.3]) is isomorphic to the principal order ideal F pL q ďX whose geometric realization is a rkpXq-ball B X with boundary }F pL q ăX }.Now [16,Lemma A.4] implies that the interior of B X includes homeomorphically into an open subset U GX of T .We have U GX X U GY " H whenever GX ‰ GY , and clearly T is the union of all U GX .We claim that this is a CW-structure on T , and we prove it by checking the conditions given in [28, §38].
First, since nerves of categories are defined as CW-complexes, the total space T is Hausdorff.The required continuous map of B X into T carrying the interior of B X to U GX and the boundary to a finite union of cells is the canonical map from the realization of the slice below GX to T (see [16,Definition A.3], where this map is called j GX ).Moreover, the closure U GX of U GX in T is a subcomplex of the geometric realization }q α pF pL qq} (i.e., the the image of j GX in [16,Definition A.3]), and since the latter is a CW-complex, it follows that any From this description follows in particular that the k-skeleton of KpA α q is the union of all U GX with rkpXq ď k.Now, X has maximal rank if and only if it is a tope and, if L has no loops, being a tope implies zepXq " H, hence X R F pL {eq for all e P E. Conversely, any X that is not of maximal rank is contained in some H e , thus YA α is exactly the pd ´1q-skeleton of KpA α q, and the complement of YA α in T is the disjoint union of all U GX where X is a tope.□ The topological tameness expressed by Lemma 5.20 allows us to apply Zaslavsky's theory of topological dissections in order to enumerate the open cells constituting the complement of a toric pseudoarrangement A α .The stepping stone is determining the poset of connected components of intersections of A α .
Proposition 5.21.The poset of connected components of intersections of A α is isomorphic to the quotient poset F pL q{G.Moreover, every intersection is topologically a torus pS 1 q d´r where r is the rank of the corresponding element in F pL q{G.
Proof.Let A :" ta 1 , . . ., a m u Ď E{G and write K " KpL q for the set of central sets of the underlying semimatroid (see Definition 3.28).Then The right-hand side union is disjoint by Lemma 5.8, and by Corollary 5.17 each of its members maps under Q G to a torus of dimension d ´rkpL rAsq.Therefore, the connected components of T a 1 X . . .X T am correspond to G-orbits of maximal elements of KpL rAsq, the index set of the r.-h.s.union.More precisely, the component }F pL {te 1 , . . ., e m uq} corresponds to such an orbit Gte 1 , . . ., e m u and has dimension rkpL q ´rkpte 1 , . . ., e m uq.Let X be the closure of te 1 , . . ., e m u in S pL q.By Lemma 5.8.( 4), we have }F pL {te 1 , . . ., e m uq} " }F pL {Xqq} and rkpXq " rkpte 1 , . . ., e m uq.Comparing with the definition of F pL q{G gives the claim.□ Theorem 5.22.Let α : G oe L denote a free and sliding action of a finitely generated free abelian group G on an FAOM L , and suppose that S pL q has a basis B P E ˚,˚.We use α to denote the induced G-semimatroid α : G oe S pL q, and let T α px, yq be the associated Tutte polynomial (see Definition A.15).Then T α px, yq computes the number of connected components of the arrangement's complement as follows: Remark 5.23.In analogy with the case of enumeration of faces of arrangements [34], by passing to contractions and using the properties of Tutte polynomials of group actions derived in [17], Theorem 5.22 leads to enumerating faces of A α of any dimension.
Proof of Theorem 5.22.This is an application of Zaslavsky's dissection theory and of the theory of group actions on semimatroids.In fact, if K is a finite CW -complex, tH e u ePE is any set of (proper) subcomplexes and tR j u jPJ is the set of connected components of the complement Kz Ť ePE H e , then [35, Lemma 1.1 and Theorem 1.2] states that ÿ jPJ κpR j q " ÿ pPP µppqκppq where P is the poset of connected components of intersections of the H e s ordered by reverse inclusion, µ is the Möbius function of P , and κ is the "combinatorial Euler number" (κpXq equals Euler characteristic of X if X is compact, and otherwise it equals the Euler characteristic of the one-point compactification, see [35, §1]).Now we can apply this to the CW-complex KpA α q and the collection of subcomplexes A α .By Lemma 5.20, all R j s are open d-cells and thus ř jPJ κpR j q " p´1q d |π 0 pT z Y A α q|.Moreover, by Proposition 5.21 the poset of connected components of intersections is F pS q{G and every GX P F pS q{G is a torus pS 1 q d´r where r is the rank of GX in F pS q{G.Thus, κpGXq ‰ 0 only if GX is a maximal element of F pS q{G and, in that case, GX is a single point, whence κpGXq " 1.In summary, we can express the desired number of connected components as where the last equality is the definition of the characteristic polynomial (see §A.1.1).Now, α is translative by Lemma 5.5, and by Theorem A.16 the characteristic polynomial of the poset F pL q{G equals p´1q d T α p1 ´t, 0q, thus the claim follows.□ 6. Open questions 6.1.Pseudoarrangements in Euclidean space.Considerable work has been devoted to the study of arrangements of (and in) general manifolds -see, e.g., the introduction of [18] for a recent account.An open problem in this context is the one discussed in [19], namely to give a topological characterization of the arrangements that can appear as realizations of a (finite) AOM.Our work suggests the following reformulation.We say that Y is a pseudohyperplane in X if there is a homeomorphism of X with R n , for some n, that carries Y to a coordinate hyperplane.In particular, XzY consists of two connected components, which we can label as the "positive" and "negative" side of Y in X.With this, call pseudoarrangement in R d any collection A " tH e u ePE of pseudohyperplanes of R n such that (1) A is finitary (every p P R d has a neighborhood that intersects only finitely many H e ), and chambers have finitely many walls; (2)  An "oriented" pseudoarrangement is a pseudoarrangement with a choice of a "positive" and "negative" side of every H e .An oriented pseudoarrangement A gives rise to a set L pA q of sign vectors on E as in §1.2.1.
(Q1) Conjecture: For every oriented pseudoarrangement A , L pA q is the set of covectors of a unique simple FAOM.Conversely, every simple FAOM arises this way.
Remark 6.1.Conditions (2) and ( 3) above define what Forge and Zaslavsky call an "affine topoplane arrangement".In the finite case, our conjecture amounts to the converse of their [19, Lemma 11] and is proved in rank 2 as [19,Theorem 13].A positive answer to (Q1) would also solve the questions stated in [19] about topology of faces and structure of intersection semilattices of pseudoarrangements.
6.2.Toric oriented matroids and pseudoarrangements.The results of Section 5 suggest the categories q α pF pL qq associated to a free and sliding action on an FAOM L as the counterpart of covector posets for toric arrangements, see Section 1.2.2 for an outline of this context.In order to fully develop an oriented matroid theory for arrangements on the torus we ask the following questions.
(Q2) Find an intrinsic axiomatic description of the class of acyclic categories that can be obtained as q α pF pL qq for a free and sliding action on an FAOM L .This framework should include for instance Aguiar and Petersen's posets of labeled necklaces [1].(Q3) Find a topological characterization of the pseudoarrangements on the torus that arise as A α " tT a u aPE{G in Section 5.3.
In this context, Pagaria [30] proposed an algebraic notion of orientable arithmetic matroid and asked whether it can be interpreted in terms of pseudoarrangements in the torus.Every toric pseudoarrangement in the sense of Definition 5.3 has an associated matroid with multiplicity (via the induced group action on the underlying semimatroid), but the multiplicity does not have to be arithmetic -for example, the one associated to the pseudoarrangement in [17,Fig. 11] is not.On the other hand, for example the multiplicity matroid underlying to the nonstretchable infinite pseudoarrangement in [17, Fig. 2, left-hand side] is arithmetic, and indeed orientable -take any orientation of the underlying uniform matroid U 2,5 -(however, of course it has not the "GCD" property from [30, §8]).
(Q4) Does every orientable arithmetic matroid as defined by Pagaria [30] arise from a toric pseudoarrangement in the sense of §5.3?
6.3.Further topological interpretation: Salvetti complexes.To every oriented matroid one can associate a cell complex that, in case the oriented matroid comes from a finite arrangement of hyperplanes, carries the homotopy type of the complement of the arrangement's complexification.This cell complex is usually called "Salvetti complex", and its cohomology is described by the underlying matroid's Orlik-Solomon algebra [20].The construction can be carried out also for infinite affine arrangements and, indeed, for every FAOM L , yielding a regular CW-complex with poset of cells M pL q.Moreover, given a group action α : G oe L , we can consider the category M pαq :" M pL q G.If α is the action by deck transformations on the periodic hyperplane arrangement that is obtained by lifting a toric arrangement to the universal cover of the d-torus, then M pαq is the category studied in [14, §2.5].In particular, the Poincaré polynomial of }M pαq} is t d T α p2 `1{t, 0q (recall Remark A.17 and see [12, Theorem 3.5.(2)]), the space }M pαq} is minimal [14, Corollary 6.10] and its cohomology algebra can be described by an Orlik-Solomon type presentation [10].Now let α be any free and sliding action of a finitely generated free abelian group G on a FAOM with a basis B Ď E ˚,˚.
(Q5) Is t |B| T α p2 `1{t, 0q the Poincaré polynomial of }M pαq}?Is }M pαq} minimal?(Q6) Is there a presentation of the cohomology algebras H ˚p}M pαq}, Qq and H ˚p}M pαq}, Zq that extends the one for the case of toric arrangements given in [10]?6.4.Beyond FOAMs.In many of our results one can observe that indeed not all axioms of FAOMs are needed.This leads to questions concerning the generalizability of the theory.We want to single out the study of finitary and general COMs and their group actions as a worthwhile endeavour.A natural problem to attack in this spirit is the following.
Since any two geometric complexes realizing the same abstract complex have homeomorphic underlying spaces, it makes topological sense to talk about "the" geometric realization of an abstract simplicial complex.The dimension of an abstract simplex σ P Σ is one less than its cardinality, which corresponds to the dimension of any geometric simplex realizing the abstract complex 2 σ .A geometric (resp.abstract) simplicial complex is called pure if all its maximal simplices (resp.its maximal elements) have the same dimension (resp.cardinality).
It is customary to call facets of a simplicial complex its nontrivial faces of maximal dimension.A facet's facet is a ridge of the given complex.
A powerful tool for the study of the topology of simplicial complexes is the theory of shellability introduced by Björner.).An abstract simplicial complex Σ pure of dimension d is called shellable the set of its maximal elements can be well-ordered in a way that, for every σ that is not the initial element in the order, σ X ´Ťσ 1 ăσ 2 σ 1 ¯is a pure, pd ´1q-dimensional complex.Shellable complexes have very restricted homotopy types (see, e.g., [7], and Remark A.5).
The link of a simplex σ P Σ is the abstract simplicial complex Lkpσq :" tτ P Σ | τ X σ " H, τ Y σ P Σu.If Σ is pure of dimension d, then Lkpσq is pure of dimension d ´|σ|.Moreover, every shelling order for Σ induces a shelling order for Lkpσq.
A.1.3.Order complexes of posets.To every partially ordered set P one can associate the family ∆pP q of all finite subsets of P that are totally ordered as sub-posets of P (such finite subsets are called chains of P ).The collection ∆pP q is called the order complex of P , and it is naturally an abstract simplicial complex, with P as vertex set.We write }P } for the geometric realization of the order complex of P .
Remark A.2. Notice that the geometric realization of a chain is a simplex with one vertex for each element of the chain.Therefore, the length of the chain corresponds to the dimension of the simplex.The length of a poset P is the maximum length of a chain in P (if this maximum exists), and thus corresponds to the dimension of }P }.If P is a ranked poset, for every p P P the dimension of }P ďp } equals the rank of p.
There are several techniques available in order to determine whether the order complex of a poset is shellable.We recall the notion of a recursive coatom ordering.Definition A.3 ( [33, §7]).Let P be a graded poset of finite length ℓ and let ă be a wellordering of the set U of coatoms of P (i.e., the elements of P covered by p 1).This ordering is a recursive coatom ordering for P if either ℓ ď 2, or ℓ ą 2 and, for every coatom a of P there is a recursive coatom ordering of P ďa in which the elements of the set Q a of coatoms of P ďa X p Ť a 1 ăa P ďa 1 q come first.
Remark A.4.Notice that, for every p P P , the set Q p is completely determined by the ordering ă (see discussion after [8, Definition 4.7.17]).
of SCAT with objects all acyclic categories.Although AC is cocomplete, its colimits do not coincide with colimits taken in SCAT.In particular, given C P ObpACq and an action of a group G on C, we write C G for the quotient object in AC.See [14, §5.1] for a study of colimits in AC and [16,Appendix A.4] for an explicit treatment of quotients by group actions.
In particular, if a group G acts on an acyclic category C then it acts on the cell complex }C} and, under certain conditions there is an isomorphism of cell complexes }C G} » }C}{G (see, e.g., [16,Corollary A.20]).
Remark A.8 (P {G and P G).A poset P can be thus considered as an object in the category POS of partially ordered sets and order preserving maps, or as an acyclic category, as described above.This distinction entails that on posets we may consider two different types of "quotients".Let a group G act on a poset P .Then on the set P {G of orbits of elements of P we can consider a binary relation ď defined by Gp ď Gq if and only if p ď gq for some g P G.Under some mild conditions, this defines a partial order relation (e.g., when P is of finite length, see [13, §2.1]) and we can speak of the quotient poset P {G.It is important to underscore that in general the poset P {G, viewed as an acyclic category, is different from the quotient category P G.
A.2. Semimatroid theory.We recall some basic definitions and set up some notations from the theory of semimatroids and geometric semilattices, and group actions thereon.We follow mainly [17], to which we refer for a more thorough treatment as well as for a brief historical sketch of the subject.
Definition A.9.A finitary semimatroid is a triple S " pE, K, rkq consisting of a (possibly infinite) set E, a non-empty simplicial complex K on E and a function rk : K Ñ N satisfying the following conditions.
(R1) If X P K, then 0 ď rkpXq ď |X|.(R2) If X, Y P K and X Ď Y, then rkpXq ď rkpY q. (R3) If X, Y P K and X Y Y P K, then rkpXq `rkpY q ě rkpX Y Y q `rkpX X Y q. (CR1) If X, Y P K and rkpXq " rkpX X Y q, then X Y Y P K. (CR2) If X, Y P K and rkpXq ă rkpY q, then X Y y P K for some y P Y ´X.
Definition A.10.A loop of a semimatroid S " pE, K, rkq is any s P E with rkpsq " 0. We call S simple if it has no loops and if rkpx, yq " 2 for all tx, yu P K with x ‰ y.Definition A.11.Let S " pS, K, rkq be a finitary semimatroid and X P K.The closure of X in K is clpXq :" tx P S | X Y x P K, rkpX Y xq " rkpXqu.
A flat of a finitary semimatroid S is a set X P K such that clpXq " X.The set of flats of S ordered by containment forms the poset of flats of S .A.2.2.Geometric semilattices.Geometric semilattices offer a poset-theoretic cryptomorphism for semimatroids, just as geometric lattices do for matroids.However, the study of geometric semilattices goes back to work of Wachs and Walker [33] that predates the introduction of semimatroids and does not restrict to the finite case.We review the definition and prove a cryptomorphism, slightly improving on [17,Theorem E].Recall that a set A of atoms of a ranked, bounded-below meet-semilattice is called independent if _A exists and its the poset-rank equals the cardinality of A. This is equivalent to saying that A is minimal under all A 1 such that _A 1 " _A.
Definition A.12 (See Theorem 2.1 in [33]).A geometric semilattice is a chain-finite ranked meet-semilattice L satisfying the following conditions.
(GSL1) Every (maximal) interval in L is a finite geometric lattice.(GSL2) For every independent set A of atoms of L and every x P L such that rk L pxq ă rk L p_Aq, there is a P A with a ę x and such that x _ a exists.
Theorem A.13.A poset L is a geometric semilattice if and only if it is isomorphic to the poset of flats of a finitary semimatroid.More precisely, there is a one-to-one correspondence between (1) Subposets L Ď P fin pEq of the poset of finite subsets of a set E ordered by inclusion, such that L is a chain-finite ranked meet-semilattice with respect to set intersection, such that (GSL2) holds and such that, for all X P L, L ďX is the (geometric) lattice of flats of a matroid.(2) Finitary semimatroids pE, K, rk L q.
The correspondence is as follows: given L as in (1) let K :" Ť XPL 2 X and for every A P K define A :" min L tX P L | X Ě Au (this is well-defined because L is a meet-semilattice).Let then rkpAq :" rk L pAq, where rk L denotes the rank function of L. Then pE, K, rk L q is a finitary semimatroid with poset of flats isomorphic to L.
Proof.The proof that the poset of flats of a semimatroid is a geometric semilattice can be found in [17] (Notice also Remark 5.3.there).Now let L Ď P fin pEq be as in the claim and consider the triple pE, K, rk L q.The family K is by definition a simplicial complex on E. Since L is chain-finite, the function rk L , that measures length of maximal chains in intervals, takes value in N, and so does rk.Now, (R1), (R2) and (R3) are statements about the value of rk L on intervals of L (specifically: (R1) on the interval L ďX , (R2) on the interval L ďY , (R3) on the interval L ďXYY ).By assumption these intervals are geometric lattices of flats of matroids, and hence the corresponding restriction of rk is a matroid rank function (on X, resp.Y , X Y Y ) -in particular, rk satisfies (R1), (R2), (R3), see, e.g., [29].
For (CR1) Let X, Y P K. Trivially X X Y Ď X, and so rkpXq " rkpX X Y q implies X X Y " X.Now, X X Y Ď Y and thus X " Y .In particular, L contains an element (e.g., X) that contains X Y Y , whence X Y Y P K.
(SE " ) X, Y P O, X " Y ùñ @e P SpX, Y q : I " e pX, Y q ‰ H.It is easy to see that (also for arbitrary ground sets E) the above definition is equivalent to requiring (0), (FS), and (SE), see [4].
A fundamental theorem in oriented matroid theory says that O is an OM if and only if it is the set of sign vectors of an arrangement of pseudospheres (i.e., a set of centrally symmetric, tame embeddings of codimension-1 spheres in a sphere, with some conditions on their intersections, see [8, §5.1]).In this interpretation, finite affine oriented matroids appear as sign vectors of the arrangement induced on one of the the open hemispheres obtained by removing one pseudosphere from the ambient sphere.
Proposition A.19 ( [6, Theorem 2.1]).Let E be a finite set.A subset L Ď t`, ´, 0u E is the set of covectors of a finite AOM if and only if there is an oriented matroid O Ď t`, ´, 0u EZtgu such that L " tX P O | Xpgq " `u.
Remark A.20.Notice that in the setting of Proposition A.19 we have that F pL q is an order filter in F pOq.In particular, it is ranked (see also [8,Proposition 4.5.3]).
Borrowing some terminology from the theory of arrangements, we say that O is obtained by coning L with respect to the new element g.It is noteworthy that the set of sign vectors in O is fully determined by L as in the following remark.
Remark A.21 ( [6, Lemma 2.2]).Let L be a finite AOM on the ground set E. Let O be the set of covectors of the finite OM obtained by coning L with respect to an additional element g R E. Then Oz pL Y ´L q " tN P t`, ´, 0u EYg | N pgq " 0, p˘N zgq ˝L Ď L u. Proof.Let O be the set of covectors of the finite OM obtained by coning L with respect to an additional element g R E. Let Y 1 be bounded in L rAs and let Y be such that Y |A " Y 1 .By definition, Y is bounded if and only if O ďY Ď L .In view of Remark A.21, this means that there is no X P O ďY such that p˘Xq ˝L Ď L .By way of contradiction suppose that such an X exists and consider X 1 :" X |A .Then, for every Z P L rAs we can choose Z P L such that Z |A " Z and then in particular, p˘X 1 q ˝Z " `p˘Xq ˝Z˘| A P L rAs (since p˘Xq ˝Z P L ).Since obviously X 1 ď Y 1 in L rAs, again by Remark A.21 we conclude that Y 1 is unbounded in L rAs, a contradiction.□

4 (Figure 1 .
Figure 1.An arrangement of hyperplanes in R 2 with some cells labeled by the respective sign vector.

Figure 3 .
Figure 3.The quotient of the poset of covectors of the pseudoarrangement in Figure 1.2.1 is the face category of a (pseudo)arrangement in the 2-dimensional torus (e.g., obtained by identifying opposite sides of the shaded rectangle), whose cells are counted by the Tutte polynomial of the group action on the underlying semimatroid.

Figure 4 .
Figure 4. Covector axioms illustrated in the example from Figure 1.

Lemma 2 . 15 .
COMs are closed under minors, i.e., the properties (FS) and (SE) are closed under deletion and contraction.

Figure 5 .
Figure 5.An example of an infinite (but periodic) pseudoline arrangement.The point highlighted by a hollow bullet corresponds to the covector X B described in Example 3.31.

(Proposition 2 . 27 .
BR1) ra, b, cs implies that a, b, c are distinct.(BR2) rωpaq, ωpbq, ωpcqs holds for some permutation ω of ta, b, cu.(BR3) ra, b, cs implies rc, b, as (BR4) ra, b, cs and ra, c, bs are mutually exclusive (BR5) ra, b, cs implies at least one of ra, b, xs and rx, b, cs, whenever x R ta, b, cu.Let π Ď E denote a parallelism class of a simple AOM.The ternary relation on π defined by

( 1 )
|Y | ď 2, and (2) the boundary of }F pL q} is generated by all chains of the form ω 1 with |Y | " 1. Proof.For (1) first notice that by (Z) we have |Y | ă 8.By Lemma 3.11 we can find a finite convex set C such that ω Y Y Ď L rE C s. Now the claim follows with Lemma 3.14 from the corresponding property in the finite affine oriented matroid L rE C s (see [8, Theorem 4.1.14],

Figure 8 .
Figure 8. Three examples of polyhedral complexes realizing }F pL q}.From top left clockwise, they correspond to the FAOMs in Figure 1, Figure 6.(c) and Figure 6.(a).The resulting cell complexes (whose barycentric subdivision is }F pL q}) are shaded gray: they are homeomorphic to a closed disk, R 2 and R 1 ˆr0, 1s, respectively.

Definition 4 . 1 (Example 4 . 2 .
Basis frame).Let B be a basis of the underlying semimatroid S pL q.The associated basis frame is r B :" ď bPB πpbq Ď E, the union of all parallelism classes of elements of B. Recall from Example 3.31 that the pseudolines labeled a 0 , b 0 in Figure 5 form a basis B " ta 0 , b 0 u of the associated FAOM.The pseudolines corresponding to the basis frame r B for are drawn in a bold stroke in Figure 5.

r Bs L rB ˝Y teus L rB ˝s ϕ Figure 11 .
Figure 11.A picture for the setup of the proof of Lemma 4.11.

Definition A. 22 .
A bounded face of a finite AOM L that arises from an OM O is any X P L such that F pL q ďX " F pOq ďX , compare [8, Definition 4.5.1].Lemma A.23.Let L be a finite AOM on the ground set E, let A Ď E and let Y 1 be a bounded face of L rAs.Then every Y P L such that Y |A " Y 1 is a bounded face of L .
2. 1.3.Acknowledgements.Emanuele Delucchi has been partially supported by the Swiss National Science Foundation professorship grant PP00P2 150552/1.Kolja Knauer was supported by the Spanish State Research Agency through grants RYC-2017-22701, PID2022-137283NB-C22 and the Severo Ochoa and María de Maeztu Program for Centers and Units of Excellence in R&D (CEX2020-001084-M) by the French Agence nationale de la recherche through project ANR-17-CE40-0015.
Therefore, ZzA P L {A and it satisfies (SE).
if Y " Ş ePA H e is the intersection of some of the pseudohyperplanes and e P E is such that H e X Y ‰ H, then either Y Ď H e or Y X H e is a pseudohyperplane in Y ; (3) the intersection of any family of elements of A , when nonempty, is clean in the sense of Bott (i.e., every point of R d has an open neighborhood U with a homeomorphism U Ñ R n that sends the intersections of the H e with U to a finite arrangement of linear hyperplanes, see [18, Section 3] for a precise statement); (4) parallelism (H e }H f if and only if H e X H f " H) is an equivalence relation.