Diskrete Mathematik und Optimierung

Diskrete Mathematik und Optimierung Winfried Hochstättler, Robert Nickel: Joins of Oriented Matroids Technical Report feu-dmo009.07 Contact: winfried.hochstaettler@fernuni-hagen.de robert.nickel@fernuni-hagen.de FernUniversität in Hagen Fakultät für Mathematik und Informatik Lehrstuhl für Diskrete Mathematik und Optimierung D 58084 Hagen

2000 Mathematics Subject Classification: 52C40, 52B40, 05B35 Keywords: Oriented Matroids, Matroid Constructions, Matroid Joins, Amalgams, Series and Parallel Connection

Joins of Oriented Matroids Winfried Hochstättler, Robert Nickel November 20, 2007 Abstract We define series/parallel/2-sum connection of two oriented matroids in terms of various axiom systems and an oriented modular join and sum operation by means of signed cocircuits and covectors. 1 Introduction Parallel, series, 2-sum, and generalized parallel connection of two (non-oriented) matroids are well known operations in matroid theory (see Brylawski [3, Chapter 7]). Although a generalization to oriented matroids is natural and meaningful, it appeared in the literature only partially and very recently in independent papers [5] and [8]. Dong [5] defined a parallel connection O 1 P O 2 in terms of covectors and Hochstättler and Nickel [8] defined a 2-sum O 1 2 O 2 via the sets of circuits. The purpose of this paper is to prove that these definitions are compatible, i. e. O 1 2 O 2 = (O 1 P O 2 ) \ g and O 1 2 O 2 = (O 1 S O 2 ) / g. and to work out how 2-sum, series and parallel connection (see Figure 1) act on the different cryptomorphic axiom systems of oriented matroids. To take a leaf out of Brylawski s book [3], we will formulate these operations in terms of circuits, vectors, cocircuits, covectors, chirotopes, and the convex closure operator. Finally, we define a modular join and a modular sum of two oriented matroids in terms of cocircuits and covectors enabling us to glue together oriented matroids at suitable flats of arbitrary dimension. 2 Definitions and Notation We assume familiarity with oriented matroid theory and freely use the notation defined in [1]. Let O 1, O 2 be two oriented matroids of rank r 1 resp. r 2 and element sets E 1, E 2. Let furthermore χ i, C i, V i be the chirotope, set of signed circuits resp. set of signed vectors of O i and D i and L i the sets of signed cocircuits resp. covectors for i = 1, 2. We denote by M i := O i the underlying 1

g g g g P S 2 Figure 1: parallel, series, and 2-sum connection of two digraphs. matroid of O i and by B i, F i, D i, r i, resp. cl i its set of bases, flats, cocircuits, its rank function resp. matroid closure operator. In the next section we will introduce the parallel, series, and 2-sum connection of O 1 and O 2 denoted by O 1 P O 2 (resp. S, 2 ). Occasionally, we will use as a placeholder for P, S, 2. If we e. g. write C 1 C 2 or C we actually mean C(O 1 O 2 ) (resp. B, V, L, D). For an arbitrary signed subset F let F = {e : F (e) 0} be the support of F, z(f ) := E \ F its zero set, and F g := (F + ({g} F ), F ({g} F )) the reorientation on g, where denotes the symmetric difference. For a family of signed subsets F let F = {F F F} be the family of supports and g F := { F g : F F} the reorientation on g. If T E we write F T := (F + T, F T ). For two signed subsets F 1, F 2 of E 1, E 2 let F 1 F 2 denote the composition defined by { F 1 (e) if F 1 (e) 0 (F 1 F 2 ) e := F 2 (e) otherwise. Let T E and F 1 T = F 2 T. Then we furthermore define the nonstandard operation composition with T -deletion by For two families F i 2 Ei, i = 1, 2 let F 1 \ TF 2 := (F 1 F 2 ) \ T. F 1 F 2 := {F 1 F 2 F 1 F 1, F 2 F 2 } and F 1 T F 2 := {F 1 F 2 F 1 F 1, F 2 F 2, F 1 T = F 2 T } F 1 \ T F 2 := {F 1 \ TF 2 F 1 F 1, F 2 F 2, F 1 T = F 2 T }. If T = {g} for some g E we write X 1 \ gx 2, F 1 g F 2, resp. F 1 \ gf 2 instead. If 2

F is a family of signed subsets of E and T E, then we define F \T := {F F F T = } F T := {F F F T } and, again, write F \g resp. F g instead of F \{g} resp. F {g}. We need similar definitions for families X of unsigned subsets of a finite set E as well: X \T := {X X : X T = } X T := X \ X \T X 1 X 1 := {X Y : X X 1, Y X 2 } X 1 T X 1 := {X Y : X X 1, Y X 2, X T = Y T } X 1 \ TX 1 := {(X Y ) \ T : X X 1, Y X 2, X T = Y T }. To connect two sets of bases we furthermore define B 1 gb 2 := {(B 1 B 2 ) \ g B 1 B 1, B 2 B 2, g B 1 B 2 }. The Convex Closure Operator The convex closure operator was defined by Folkman and Lawrence [6] and uses a slightly different notion of an oriented matroid. There, the oriented matroid acts on a set E with an involution : E E. Actually, any element e E is contained in E together with its copy e. With this notation, analog circuit axioms (see [4, Theorem 3]) characterize entire reorientation classes of oriented matroids but not single orientations. We use a modified version of this notation that appears in [7] and makes the convex closure more compatible with standard notation. Let ±E := {+e, e e E} and for some A ±E let σa τ := {σe τe A} for σ, τ {+, } (partitioning E = +E E this way chooses a particular orientation). We will always refer to an element of ±E together with its sign. For some signed subset F = (F +, F ) of E and a set A ±E we write F A if +F + F A and we abbreviate A \ e := A \ {+e, e}. Büchi and Fenton [4] explicitly state the definition of an oriented matroid in terms of a convex closure operator (see also [1, Exercise 3.11]): Definition 1. A function conv : 2 ±E 2 ±E is called the convex closure operator of an oriented matroid if it satisfies (CV1) conv( ) =. (CV2) A conv(a) = conv(conv(a)). (CV3) A B conv(a) conv(b). (CV4) conv( A) = conv(a). (CV5) σe conv(a σe) σe conv(a). (CV6) σe conv(a τf) and σe conv(a) τf conv(a \ τf σe). 3

Figure 2: In case of a digraph, an arc f is in conv(a) with the respective sign iff there is a dipath in A connecting its endpoints. Bold arrows indicate that an arc is contained in A with the respective sign and thin arrows beside an arc indicate that the arc is in conv(a) \ A. It is shown in [4, 7] that the convex closure operator yields a cryptomorphic characterization of oriented matroids. In particular C conv := {(A +, A ) A = +A + A is a minimal nonempty set with A conv(a)} satisfies the circuit axioms of an oriented matroid O conv and conv(a) = A {τf ±E C C conv : τf C A τf}. 3 Series, Parallel and 2-Sum Connection In this section let E 1 E 2 = {g}. The definition of the connections of O 1 and O 2 along g will be given in terms of circuits, vectors, cocircuits, covectors, signed bases (chirotopes), and the convex closure operator. 3.1 Circuits and Vectors Hochstättler and Nickel [8] introduced a 2-sum O 1 2 O 2 via the sets of signed circuits: C 1 2 C 2 := C \g 1 C \g 2 (C 1 \ g g C 2 ). At first we derive compatible series and parallel connection. Proposition 2. C 1 P C 2 := C 1 C 2 (C 1 \ g g C 2 ) C 1 S C 2 := C \g 1 C\g 2 (C 1 g g C 2 ) Proof. It is straightforward to verify that C 1 C 2 satisfy the circuit axioms of oriented matroid theory. We give the details only for C 1 S C 2 (see also [8, 4

Proposition 6.1]). Obviously, C S is antisymmetric and its support forms a clutter. It suffices to verify oriented circuit elimination for C 1, C 2 C S with e C 1 + C 2. This is straightforward if at least one of C i is in C \g i. Thus, let C 1 = C 11 C 12 and C 2 = C 21 C 22 for circuits C ij C g j (i = 1, 2) and wlog. e C 11 + C 21 E 1. If C 11 (g) = C 21 (g) then also C 12 (g) = C 22 (g) and eliminating g between C 12 and C 22 we find C 3 C \g 2 C S as required. Otherwise, C 11 (g) = C 21 (g) and hence, e g. Using circuit elimination in O 1 by fixing g we find a circuit C 31 with C 31 (C 11 + C+ 21 )\e and C 31 (C 11 C 21 )\e. Now, C 3 := C 31 C 12 is a circuit as required. Corollary 3. C 1 2 C 2 = (C 1 P C 2 ) \ g = (C 1 S C 2 ) / g. We will now determine V from C. Corollary 4. V 1 P V 2 = V 1 V 2 V 2 V 1 (V 1 \ g g V 2 ) V 1 S V 2 = V \g 1 V\g 2 (V 1 g g V 2 ) V 1 2 V 2 = V \g 1 V\g 2 (V 1 \ g g V 2 ) Proof. We work out details of the parallel connection only, the other cases being similar. is trivial. For let V V P. We know from Proposition 3.1 that V P is the set of vectors of an oriented matroid. Therefore (see [1, Corollary 3.7.6]), V is a conformal composition of circuits, i. e. V = C 1... C k with C i (e)c j (e) {0, +} for all e E. If g V then C i C \g 1 C \g 2 (C 1 \ g C g 2 ) and (since V \g 1 V \g 2 V 1 V 2 ) V V 1 V 2 (V 1 \ g V g 2 ). Otherwise, assume wlog. V (g) = +. Replacing each circuit C j (C 1 \ g C g 2 ) by C i1 j Ci2 j with i 1, i 2 {1, 2}, i 1 i 2 such that (C i1 j Ci2 j )(g) = +, we may assume that V = D 1... D l with D i C 1 C 2. Note that this is conformal up to g. Let i be the first index with D i (g) 0 (hence, D i (g) = +) and wlog. D i C g 1. Then we can rearrange the circuits D j so that V = (D i1... D im ) (D j1... D jm ) with D il C 1 (l {1,..., m}) and D jl C 2 (l {1,..., m }). Hence, V = V 1 V 2 V 1 V 2. 3.2 Covectors and Cocircuits The parallel connection in terms of covectors as proposed by Dong [5] L 1 P L 2 := L \g 1 L\g 2 (L 1 g L 2 ) is compatible with the 2-sum of Hochstättler and Nickel [8]: Proposition 5. L 1 P L 2 is the set of covectors of C 1 P C 2. 5

Proof. Dong [5, Proposition 4.2] proved for affine oriented matroids that O 1 P O 2 is the parallel connection of O 1 and O 2, but the proof does not use affinity. To show that the signs are correct, it remains to verify that each covector is orthogonal to each circuit. Obviously, L \g 1 L\g 2 is orthogonal to C P and C i is orthogonal to L P for i = 1, 2. Now let C = C 1 \ gc 2 C 1 \ g C g 2 and L = L 1 L 2 L 1 g L 2, wlog. C(g) = L(g) = +, and assume for a contradiction that L C. It follows that C 1 (e)l 1 (e) {σ, 0} e E 1 \ g and C 2 (e)l 2 (e) {σ, 0} e E 2 \ g. for some σ {+, }. Because of L i C i for i = 1, 2 we must have + = C 1 (g)l 1 (g) = σ = C 2 (g)l 2 (g) =. As a direct consequence we can determine the set of signed covectors of the 2-sum by deletion of g (resp. contraction of g in the dual). Corollary 6. L 1 2 L 2 := L \g 1 L\g 2 (L 1 \ gl 2 ) is the set of signed covectors of O 1 2 O 2. Corollary 4 together with Proposition 5 yield a nice analog to the duality of series and parallel connection (i. e. (O 1 P O 2 ) = O 1 S O 2 ). In case of oriented matroids the sign of g switches under this duality. For that purpose let g O i be the reorientation of O i with respect to g. Then Corollary 7. (O 1 P O 2 ) = O 1 S g O 2 and (O 1 S O 2 ) = O 1 P g O 2 and, as a direct consequence, we get for the covectors of the series connection Corollary 8. The set of covectors of C 1 S C 2 is given by L 1 S L 2 = L 1 g L 2 g L 2 L 1 (L 1 \ gl 2 ). Considering that cocircuits are covectors of minimal support we get Corollary 9. D 1 P D 2 = D \g 1 D \g 2 (D 1 g D 2 ) D 1 S D 2 = D 1 D g 2 (D 1 \ gd 2 ) D 1 2 D 2 = D \g 1 D \g 2 (D 1 \ gd 2 ) 6

3.3 Chirotopes The set of bases of a matroid which is the parallel connection of matroids with bases B 1 and B 2 (see Brylawski [3]) is given by B 1 P B 2 = B g 1 Bg 2 B\g 1 gb g 2 Bg 1 gb \g 2 = B g 1 Bg 2 B 1 gb 2. The signed analog of bases are called chirotopes. Recall that χ i : B i {+,, 0} are the chirotopes of O i. From now on assume that bases are always given with an ordering. If B = B 1 B 2, B i B i then let B i have the ordering induced by B. Lemma 10. The function χ P : B P {+,, 0} defined by χ P (B) := χ 1 (B 1 )χ 2 (B 2 ), where B B P and therefore, B = B 1 B 2 with g B 1 B 2 resp. B = (B 1 B 2 ) \ g with g B i for exactly one i {1, 2} for some B 1 B 1, B 2 B 2, is the chirotope of O 1 P O 2 Proof. We prove that χ P defined as above is a proper basis orientation of B P (see e. g. [1, Definition 3.5.1]). Let B, B B P with B B = r 1 + r 2 2 and e 1 := B \ B e 2 := B \ B X 1 := (B \ e 1 ) E 1 = (B \ e 2 ) E 1 X 2 := (B \ e 1 ) E 2 = (B \ e 2 ) E 2 and thus B = e 1 X 1 X 2 and B = e 2 X 1 X 2 and let C be the (up to sign reversal unique) circuit in {e 1, e 2 } X 1 X 2. We have to show that χ P (B ) = C(e 1 )C(e 2 )χ P (B). We consider the following cases: X 1 X 2 = {g} : Hence, B, B B g 1 Bg 2. Since X 1 X 2 = 1 and X 1 X 2 = r 1 + r 2 2 we may assume wlog. that X 2 = r 2. Thus, X 2 B g 2 and since X 2 e i is independent for i = 1, 2, we must have e 1, e 2 E 1 \ g. The claim follows since χ 1 is a proper basis orientation of B 1. X 1 X 2 = : If e 1, e 2 E i for some i, again the claim follows since χ i is a proper basis orientation of B i. Otherwise, let wlog. e i E i and B B \g 1 gb g 2 implying that X 1 e 1 and X 2 g are bases. Then B B P implies that B must be in B g 1 gb \g 2 and therefore, X 1 g and X 2 e 2 must be bases as well. Since C must contain e 1 and e 2, we have that 7

C = C 1 g C 2 for circuits C 1 C g 1 and C 2 C g 2 with C 1(g) = C 2 (g). Hence, we can exchange e 1 with g and on the other side g with e 2 and obtain χ P (B ) = χ 1 (X 1 g)χ 2 (X 2 e 2 ) = ( C 1 (g)c 1 (e 1 )χ 1 (X 1 e 1 )) ( C 2 (e 2 )C 2 (g)χ 2 (X 2 g)) C 1(g)C 2(g)= = C 1 (e 1 )C 2 (e 2 )χ 1 (X 1 e 1 )χ 2 (X 2 g) = C(e 1 )C(e 2 )χ P (B). and Deleting g from O 1 P O 2 we conclude that (c. f. [3]) B 1 2 B 2 = B \g 1 gb g 2 Bg 1 gb \g 2 Corollary 11. The chirotope χ 2 of O 1 2 O 2 is given by χ 2 (B) = χ 1 (B 1 )χ 2 (B 2 ), where B B 2 exactly one i. with B = (B 1 B 2 ) \ g for B i B i, i = 1, 2 and g B i for The basis orientation of the series connection is governed by the set of bases of the underlying matroid (see [3]) as well: B 1 S B 2 = B \g 1 B\g 2 B \g 1 Bg 2 B g 1 B\g 2. Proposition 12. The chirotope χ S of O 1 S O 2 is given by χ S (B) = χ 1 (B 1 )χ 2 (B 2 ), where B B S and therefore, B = B 1 B 2 for some B 1 B 1, B 2 B 2. Proof. If e 1, e 2 E i for some i, the claim follows since χ i is a proper basis orientation of O i. Let B, B B S with B B = r 1 + r 2 2 and e 1, e 2, X 1, X 2, C be defined as in Lemma 10. If g X 1 X 2, then we conclude e 1, e 2 E i for i {1, 2} as follows. If g {e 1, e 2 }, this is immediate. Otherwise, if e 1 = g, we may wlog. assume that X 1 g and X 2 are bases and hence, e 1, e 2 E 1. The only remaining case is g X 1 X 2 and wlog. e i E i for i = 1, 2. By the definition of the X i we have g X 1 X 2. Because B B \g 1 B\g 2 implies e 1, e 2 E i for some i, we may, by symmetry, assume B B \g 1 Bg 2 and 8

B B g 1 B\g 2. Thus, X 1, X 1 \ g e 1, X 2, and X 2 \ g e 2 are bases of O 1 resp. O 2. Hence, χ S (B ) = χ 1 (X 1 )χ 2 (X 2 \ g e 2 ) = ( C 1 (g)c 1 (e 1 )χ(x 1 \ g e 1 )) ( C 2 (e 2 )C 2 (g)χ 2 (X 2 )) C 1(g)C 2(g)= = C 1 (e 1 )C 2 (e 2 )χ 1 (X 1 \ g e 1 )χ 2 (X 2 ) = C(e 1 )C(e 2 )χ S (B). Remark 13. Björner et al. [1, Section 7.6] considered two special cases of oriented matroid union, i. e. disjoint ground sets (direct sum) and equal ground sets. As (unoriented) series connection is a special case of matroid union (see e. g. [9, Proposition 12.3.6]), Proposition 12 is another special case of oriented matroid union. 3.4 Convex Closure We are now going to determine the convex hull operators conv of O from the sets of circuits. For O i let conv i := conv Ci (i = 1, 2) and conv := conv C1 C 2. We identify conv i (A) with conv i (A E i ) for arbitrary sets A (i = 1, 2) simplifying notation. First we consider the parallel connection: Theorem 14. Let G 1 := {+g, g} conv 2 (A) and G 2 := {+g, g} conv 1 ( A). Then conv P (A) = conv 1 (A G 1 ) conv 2 (A G 2 ). Proof. Let τf conv P (A) for some τ {+, } and wlog. τ = +. The case +f A is trivial and otherwise, there is a circuit C C P such that f C A f. If C C i for i = 1 or 2, then +f conv i (A) conv i (A G i ). Otherwise, there is a circuit C = C 1 \ gc 2 C 1 \ g C g 2. If f C 1 and C 1 (g) = σ, we have the implication f C 1 C 2 \ g A σg conv 2 (A) σg G 1 f C 1 A G 1 f +f conv 1 (A G 1 ). An analogous argument verifies f C 2 +f conv 2 (A G 2 ). 9

f g f g Figure 3: f is contained in conv P (A) but not in conv 1 (A) Let τf conv 1 (A G 1 ). Again we may assume τ = +. If +f conv i (A) or G i = then +f conv P (A) follows immediately. If +f conv 1 (A G 1 ) \ conv 1 (A) and σg G 1 then there is a circuit C 1 C g 1, C 1(g) = σ so that f C 1 A G 1 f and a circuit C 2 C g 2, C 2(g) = σ, satisfying C 2 A σg. It follows that f C 1 \ gc 2 A f and therefore, +f conv P (A). The case τf conv 2 (A G 2 ) is analogous. In Figure 3 you see an example for the convex closure operator of the parallel connection where an edge f is contained in the convex closure of the parallel connection but not in conv i (A). Red arrows indicate that an arc is in the convex closure of A with the respective sign while bold red arcs are the elements of A. We will derive the convex closure operator of the 2-sum O 1 2 O 2 from the operator of the series connection. Theorem 15. Let G 1 := {+g, g} A conv 2 (A \ g) G 2 := {+g, g} A conv 1 ( A \ g). Then conv S (A) = A conv 1 (A \ g G 1 ) conv 2 (A \ g G 2 ). Proof. Let τf conv S (A). If τf A or τf conv i (A \ g) for i = 1 or 2 the claim is true. It remains to consider τf conv S (A) \ (A conv 1 (A \ g) conv 2 (A \ g)) and wlog. τ = +. Then there is a circuit C = C 1 C 2 C 1 g g C 2 (wlog. C(g) = +) satisfying f C A f. It follows that +g A and hence, f conv 1 (A \ g G 1 ). If f C 1, then g C 2 A g and hence, +g conv 2 (A \ g). Note that G 1 = {+g, g} A conv 2 ( A \ g) G 2 = {+g, g} A conv 1 (A \ g). and Hence, the case f C 2 follows by symmetry, reorienting g. 10

f g f g Figure 4: f is contained in conv 1 (A) but not in conv S (A) Let τf conv 1 (A \ g G 1 ). The cases τf A conv 1 (A \ g) and G 1 = are trivial. Thus, τf conv 1 (A \ g G 1 ) \ (A conv 1 (A \ g)), wlog. τ = +, and σg G 1. We consider the case f {±g} first. Hence, there is a circuit C 1 C g 1 with C 1(g) = σ and f C 1 A \ g G 1 f. Since σg G 1 conv 2 (A \ g), we also have a circuit C 2 C g 2 satisfying C 2 (g) = σ and C 2 A σg. It follows that f C 1 C 2 A f and +f conv S (A). Now let f = g and hence, +g conv 1 (A \ g G 1 ) \ conv 1 (A \ g). Then +g G 1 A and hence, +g conv S (A) completing the proof. Figure 4 shows how arcs can be contained in the convex closure of one of the graphs but not in the convex closure of their series connection if g conv i (A\g). By contraction we obtain the result for the 2-sum. Corollary 16. Let G 1 := {+g, g} conv 2 (A \ g) G 2 := {+g, g} conv 1 ( A \ g). Then conv 2 (A) = conv S (A)\g = (conv 1 (A \ g G 1 ) conv 2 (A \ g G 2 ))\g. 4 Generalized Parallel Connection, Modular Join, and Modular Sum During this section O 1, O 2 are oriented matroids on the ground sets E 1 resp. E 2 such that E 1 E 2 = T and O 1 [T ] = O 2 [T ]. The underlying matroids are M i := O i and have the set of flats F i, rank function r i and matroid closure operator cl i for i = 1, 2. 11

Definition 17. Let F denote the family of flats of a matroid M with rank function r. We call two flats X, Y F a modular pair if r(x) + r(y ) = r(x Y ) + r(x Y ). A flat T is modular if for all X F X, T is a modular pair of flats. We will introduce the modular join of O 1 and O 2 as an oriented version of a special case of the generalized parallel connection from matroid theory (see e. g. [3]). First we review the basics of the generalized parallel connection from matroid theory including some seemingly new observations. Proposition 18 ([3]). If T F 1 is a modular flat of M 1 and T F 2 then the set F T := {F : F E i F i for i = 1, 2} is the set of flats of a matroid, called the generalized parallel connection of M 1 and M 2 denoted by M T. Remark 19. If T is not a flat in M 2 then one can extend M 1 by the elements cl 2 (T )\T via the modular cut {T } yielding a matroid ˆM 1 in which ˆT := cl 2 (T ) is a modular flat. The generalized parallel connection of M 1 and M 2 is defined to be the generalized parallel connection of ˆM1 and M 2 with respect to the common flat ˆT. For details on modular cuts and single element extensions we refer the reader to [9]. The rank function of the generalized parallel connection M T the following proposition. is given by Proposition 20 ([2, Proposition 5.5]). If r T, r 1, r 2 are the rank functions of M T, M 1, M 2 respectively, then for any F F T we have r T (F ) = r 1 (F E 1 ) + r 2 (F E 2 ) r 1 (F T ). As a direct consequence, the rank of the generalized parallel connection is rank(o 1 ) + rank(o 2 ) r T (T ). Proposition 21 ([2, Proposition 5.10]). E 1 F T T F T T F 2. Hence under the above assumption, E 1, E 2, and T are flats of M T. From now on we, additionally, assume that T is a common modular flat of M 1 and M 2. As a preparatory step to defining the modular join for oriented matroids, first we derive the modular join of two matroids M 1 and M 2 in terms of its cocircuits. Proposition 22. The set of cocircuits of the modular join M T = M 1 T M 2 is D T = D \T 1 D \T 2 (D T 1 T D T 2 ). 12

Proof. By Proposition 20 and since T is a modular flat in M 1 and M 2, we have for any flat H F T and H i := H E i, i = 1, 2 and by symmetry r T (H) = r T (H 1 ) + r T (H 2 ) r T (H 1 H 2 ) = r T (H 1 ) r T (H 1 T ) + r T (H 2 ) = r T (H 1 T ) r T (T ) + r T (H 2 ), r T (H) = r T (H 1 ) + r T (H 2 T ) r T (T ). Again by Proposition 20, r T equals r i when restricted to E i. Hence, a closed set H F T is a hyperplane (i. e. has rank r 1 + r 2 r T (T ) 1) if and only if r 1 + r 2 1 = r 1 (H 1 T ) + r 2 (H 2 ) = r 1 (H 1 ) + r 2 (H 2 T ), meaning that exactly one of the following cases applies: (1) H 1 = E 1 and H 2 is a hyperplane of M 2 completely containing T, (2) H 2 = E 2 and H 1 is a hyperplane of M 1 completely containing T, (3) H i are hyperplanes in M i for i = 1, 2 which do not contain T completely. In case (1) resp. (2) H is a hyperplane whose complement is a cocircuit in M 2 resp. M 1 and in case (3) H 1 and H 2 are complements of cocircuits D 1, D 2 with D 1 T = D 2 T. We are now aiming to define an oriented modular join with respect to a common modular flat T as an oriented analogue of Proposition 22. We will prove that this is well defined in Theorem 25 and start with some observations. Proposition 23. Let T be a modular flat of a matroid M and C a cocircuit such that C T {, T }. Then C T is a cocircuit of M[T ]. Proof. Let r be the rank function of M. By modularity, r(z(c) T ) = r(z(c)) + r(t ) r(z(c) T ) = rank(m) 1 + r(t ) rank(m) = r(t ) 1. The following observation will be crucial for an inductive proof of the correctness of our join operation. 13

Lemma 24. Let O 1 and O 2 be simple oriented matroids on the ground sets E 1 E 2 = T such that O 1 [T ] = O 2 [T ] and T is a common modular flat of rank 2. Then D T := D \T 1 D \T 2 (D1 T T D2 T ) is the family of cocircuits of an oriented matroid. Proof. Wlog. O is reoriented such that every X D restricted to the elements of T has one of the sign patterns 0... 0,... 0 +... +, or +... + 0... wrt. a fixed linear ordering of the elements of T. Let C D be elements of D T such that e C + D. We proceed by case study: (1) e T : Then C = C 1 C 2, D = D 1 D 2 D1 T T D2 T. If C T = D T elimination between C 1 and D 1 yields a cocircuit F 1 D \T 1 such that F1 σ (C1 σ D1 σ ) \ e for σ = +,. Otherwise, C T D T and there is some f T ((C 1 + \D 1 ) (C 1 \D+ 1 )) and we can perform strong cocircuit elimination between C i, D i for i = 1, 2 with respect to e by fixing f which yields cocircuits F i Di T satisfying F 1 T = F 2 T since T is a modular line. Hence, F 1 F 2 D1 T T D2 T is a cocircuit as required. (2) Wlog. e E 1 \T : Let F 1 D 1 be a cocircuit satisfying F1 σ (C1 σ D1 σ )\e. We are done if F 1 D \T 1. Otherwise, let f C, f D, f F be the unique elements in z(c 1 ) T, z(d 1 ) T resp. z(f 1 ) T. (i) f F = f C = f D : Then F 1 C 2 or F 1 D 2 is a cocircuit as required. (ii) f F = f C f D : Assume F 1 T = C 1 T. Then F 1 T D 1 T, a contradiction, as F 1 (f D ) 0. Hence, F 1 T = C 1 T and F 1 C 2 is a cocircuit as required. (iii) {f C, f D, f F } = 3: By cocircuit elimination we necessarily must have C(f F ) = D(f F ) 0. We eliminate f F between C 2 and D 2 in O 2 and get a cocircuit that either is in D \T 2 as required or satisfies F 2 (f F ) = 0. Since F 1 (f C ) = F 2 (f C ) = D(f C ), we must have F 1 T = F 2 T and F 1 F 2 is a cocircuit as required. Theorem 25. Let O 1, O 2 be oriented matroids with a common modular flat T = E 1 E 2. Then D T := D \T 1 D \T 2 (D T 1 T D T 2 ) is the family of signed cocircuits of an oriented matroid, called the modular join of O 1 and O 2, denoted by O T. Proof. We may wlog. assume that O 1 and O 2 are simple. We prove the theorem by induction on T. For T {0, 1} the statement corresponds to the signed cocircuits of direct sum resp. parallel connection (empty set and single edges are always modular flats). Now let T 2 and C, D D T such that C D and e C + D. If there exists some f z(c) z(d) T then C, D D T / f 14

and by inductive assumption, there is some F D T / f D T satisfying F + (C + D + ) \ e and F (C D ) \ e. We now may assume that z(c) z(d) T =. Since z(c 1 ) T is a modular flat in O 1, we have 0 = r(z(c 1 ) z(d 1 ) T ) = r((z(c 1 ) T ) (z(d 1 ) T )) = r(z(c 1 ) T ) + r(z(d 1 ) T ) r((z(c 1 ) T ) (z(d 1 ) T ) = r(t ) 2. It thus suffices to consider the case that r(t ) = 2 which was done in Lemma 24. Corollary 26. L 1 T L 2 = L \T 1 L \T 2 (L 1 T L 2 ). Please note the analogy to the parallel connection. Furthermore, it is now immediate to define the modular sum of two oriented matroids as a generalization of 2-sum. Definition 27. Let O 1, O 2 be oriented matroids on the ground sets E 1 E 2 = T so that T is a common modular flat. The modular sum O 1 \T O 2 is defined via its set of cocircuits D \T : 5 Concluding Remarks D \T := D \T 1 D \T 2 (D 1 \ TD 2 ). While parallel, series, and 2-sum connection have been studied involving the most important axiom systems, this is left open for the operations of modular join and modular sum as e. g. the set of circuits of the generalized parallel connection is not an immediate analogue to the parallel connection. Furthermore, if T contains more than one element, generalized parallel connection lacks of a meaningful dual operation which in the case of T = 1 is the series connection and corresponds to matroid union if the ground sets intersect in T. This does not hold for larger T as well. The generalized parallel connection of a matroid is well defined as soon as T is a modular flat of O 1. We leave it as an open question whether the equation in Corollary 26 yields an oriented matroid if T is not a modular flat of O 2. Note that the unoriented analogue holds. References [1] Anders Björner, Michel Las Vergnas, Bernd Sturmfels, Neil White, and Günter M. Ziegler. Oriented matroids. Cambridge University Press, Cambridge, 2nd edition, 1999. 15

[2] Thomas Brylawski. Modular constructions for combinatorial geometries. Transactions of the American Mathematical Society, 203:1 44, 1975. [3] Thomas Brylawski. Constructions. In Theory of matroids, volume 26 of Encyclopedia of Mathematics and its Applications, pages 127 223. Cambridge University Press, Cambridge, 1986. [4] J. Richard Büchi and William E. Fenton. Large convex sets in oriented matroids. Journal of Combinatorial Theory. Series B, 45(3):293 304, 1988. [5] Xun Dong. On the bounded complex of an affine oriented matroid. Discrete and Computational Geometry, 35(3):457 471, 2006. [6] Jon Folkman and Jim Lawrence. Oriented matroids. Journal of Combinatorial Theory. Series B, 25(2):199 236, 1978. [7] Winfried Hochstättler and Jaroslav Nešetřil. Antisymmetric flows in matroids. European Journal of Combinatorics, 27(7):1129 1134, 2006. [8] Winfried Hochstättler and Robert Nickel. The flow lattice of oriented matroids. Contributions to Discrete Mathematics, 2(1):68 86, 2007. [9] James G. Oxley. Matroid theory. The Clarendon Press Oxford University Press, New York, 1992. 16