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Lectures on Matroids

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... In this section, we recall some basic properties of regular matroids and regular chain groups presented in [3,4,[20][21][22][23][24]. ...
... Denote by P (N) the set of primitive chains of N and let C(M) denote the family of circuits of M = M(N). As pointed out in [22,23], ...
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A regular matroid M on a finite set E is represented by a totally unimodular matrix. The set of vectors from Z E orthogonal to rows of the matrix form a regular chain group N. Assume that ψ is a homomorphism from N into a finite additive Abelian group A and let Aψ[N] be the set of vectors g from (A−0) E , such that ∑e∈E g(e)· f(e) = ψ(f) for each f ∈ N (where · is a scalar multiplication). We show that |Aψ[N]| can be evaluated by a polynomial function of |A|. In particular, if ψ(f) = 0 for each regular matroid; regular chain group; totally unimodular matrix; homomorphism; assigning polynomial, then the corresponding assigning polynomial is the classical characteristic polynomial of M.
... We conclude the paper with a short outlook in Sect. 5, and provide a list of symbols with short summaries on p. 41. ...
... 2.3. The graph G gives rise to a graphic oriented matroid, whose dual is the cographic oriented matroid M * (G) [3,41]. ...
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The Periodic Event Scheduling Problem (PESP) is the standard mathematical tool for optimizing periodic timetables in public transport. A solution to a PESP instance consists of three parts: a periodic timetable, a periodic tension, and integer offset values. While the space of periodic tensions has received much attention in the past, we explore geometric properties of the other two components. The general aim of this paper is to establish novel connections between periodic timetabling and discrete geometry. Firstly, we study the space of feasible periodic timetables as a disjoint union of polytropes. These are polytopes that are convex both classically and in the sense of tropical geometry. We then study this decomposition and use it to outline a new heuristic for PESP, based on neighbourhood relations of the polytropes. Secondly, we recognize that the space of fractional cycle offsets is in fact a zonotope, and then study its zonotopal tilings. These are related to the hyperrectangle of fractional periodic tensions, as well as the polytropes of the periodic timetable space, and we detail their interplay. To conclude, we also use this new understanding to give tight lower bounds on the minimum width of an integral cycle basis.
... We recall properties of regular matroids presented in [1,[8][9][10][11]). For any basis B of M, D can be transformed to a form (I r |U) such that I r corresponds to B and U is totally unimodular. ...
... Mathematics 2023,11, 2570 ...
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We introduce a unifying approach for invariants of finite matroids that count mappings to a finite set. The aim of this paper is to show that if the cardinalities of mappings with fixed values on a restricted set satisfy contraction–deletion rules, then there is a relation among them that can be expressed in terms of linear algebra. In this way, we study regular chain groups, nowhere-zero flows and tensions on graphs, and acyclic and totally cyclic orientations of oriented matroids and graphs.
... Although not used in this paper, separating cocircuits have received considerable attention in the literature. See, for example, Tutte [33,34,36,37,38], Bixby and Cunningham [3], Cunningham [9], Mighton [27], and Wagner [41]. Moreover, the nonseparating cocircuit algorithm of this paper can be seen as a complement to the algorithm of Cunningham [9] (and, independently, Tamari [32]) that efficiently computes a separating cocircuit in a nonseparable binary matroid M provided that M has at least rank three and does not contain a Fano minor. ...
... Moreover, the nonseparating cocircuit algorithm of this paper can be seen as a complement to the algorithm of Cunningham [9] (and, independently, Tamari [32]) that efficiently computes a separating cocircuit in a nonseparable binary matroid M provided that M has at least rank three and does not contain a Fano minor. (Tutte [37] proved that these conditions are sufficient for the existence of such a cocircuit.) ...
... (iv) S has a row for each inequality of the form x e 0 (we refer to such rows as non-negativity rows) and a row for each inequality 1 − x e 0 for e ∈ E. We now justify Assumption (iv). One can show (using well-known facts from [24], see [1] for more details) that for each element e ∈ E at least one of the inequalities x e 0, x e 1 is facet defining for B(M ). ...
... But both matrices are 1-products, hence by Corollary 21, none of M \ e, M/e is connected. But this is in contradiction with the well-known fact that, if M is connected, then at least one of M \ e, M/e is connected (see [24]). ...
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In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra. The 1-product of matrices S1S_1, S2S_2 is a matrix whose columns are the concatenation of each column of S1S_1 with each column of S2S_2. The k-product generalizes the 1-product, by taking as input two matrices S1,S2S_1, S_2 together with k1k-1 special rows of each of those matrices, and outputting a certain composition of S1,S2S_1,S_2. Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices. In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.
... A matroid N is called an excluded minor for a class of matroids M if N / ∈ M but every proper minor of N is in M . A well-known excluded minor characterization for graphic matroids goes as follows [18], where K 5 is the complete graph on five vertices and K 3,3 is the complete bipartite graph having three vertices at each side of the bipartition: ...
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We introduce the notion of graphic cocircuits and show that a large class of regular matroids with graphic cocircuits belongs to the class of signed-graphic matroids. Moreover, we provide an algorithm which determines whether a cographic matroid with graphic cocircuits is signed-graphic or not.
... We can narrow the context of our visualization down to that of graphic matroids of planar graphs, in which the dual matroid is equivalent to taking a planar dual 4 of a fixed planar embedding of the graph. Note that while a planar graph can have multiple, non-isomorphic duals, their graphic matroids will be isomorphic [31]. Let the following be a planar embedding of G: Consider the graph G * embedded in the plane, dual to our embedding of G, such that the edge set of G * is equal to that of G. ...
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A matroid is a mathematical object that generalizes the notion of linear independence of a set of vectors to an abstract independence of sets, with applications to optimiza- tion, linear algebra, graph theory, and algebraic geometry. Matroid theorists are often concerned with representations of matroids over fields. Tutte’s seminal theorem proven in 1958 characterizes matroids representable over GF(2) by noncontainment of U2,4 as a matroid minor. In this thesis, we document a formalization of the theorem and its proof in the Lean Theorem Prover, building on its community-built mathematics library, mathlib.
... But there, according to Tutte [21], "the theory of matroids was proclaimed to the world". Edmonds arranged for Tutte to give a series of lectures on his work, and to write for publication a new exposition [20] of his main structural results. Edmonds presented his own work related to partitioning and Lehman's game. ...
... For a matroid M , recall that C(M ) denotes the set of circuits of M . A subset L of E(M ) is a Tutte-line of M if (M |L) * has rank two and has no loops [13]. As Tutte showed and is easily checked, a Tutte-line L has a partition into sets P 1 , P 2 , . . . ...
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Jim Geelen and Peter Nelson proved that, for a loopless connected binary matroid M with an odd circuit, if a largest odd circuit of M has k elements, then a largest circuit of M has at most 2k22k-2 elements. The goal of this note is to show that, when M is 3-connected, either M has a spanning circuit, or a largest circuit of M has at most 2k42k-4 elements. Moreover, the latter holds when M is regular of rank at least four.
... Higher connectivity of graphs and matroids is well explored in [14,1,15,16,17]. In this section we study the effect of splitting and element splitting operations on connected and Eulerian p -matroids. ...
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In this paper, we define generalized splitting and element splitting operations on p-matroids. p-matroids are the matroids representable over GF(p). The circuits and the bases of the new matroid are characterized in terms of circuits and bases of the original matroid, respectively. A class of n-connected p-matroids which gives n-connected p- matroids using the generalized splitting operation is also characterized. We also prove that connectivity of p-matroid is preserved under element splitting operation. Sufficient conditions to obtain Eulerian p-matroid from Eulerian p-matroid under splitting and element splitting operations are provided.
... For example, given that w 1 engages in upgrade v 0 v 1 , this condition requires that w 1 cannot engage in other upgrades. When firms' acceptable sets of workers are from a technology tree that satisfies this condition, the firms' demand type is totally unimodular since its elements form a network matrix (Tutte, 1965;see also Chapter 19.3 of Schrijver, 1986). See an illustration in Section A.3. ...
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This paper develops an integer programming approach to two‐sided many‐to‐one matching by investigating stable integral matchings of a fictitious market where each worker is divisible. We show that a stable matching exists in a discrete matching market when the firms' preference profile satisfies a total unimodularity condition that is compatible with various forms of complementarities. We provide a class of firms' preference profiles that satisfy this condition.
... De plus, il existe de nombreuses définitions intéressantes et non trivialement équivalentes (on dit qu'elles sont cryptomorphiques) des matroïdes. Cela fait des matroïdes des objets combinatoires riches et présents dans divers domaines des mathématiques, par exemple en combinatoire et physique statistique [Tut65,MP67,GGW06,SS14,Piq19b], en topologie [Mac91,GM92], ou en géométrie algébrique [Mnë88,Laf03,BB03,BL18,Stu02]. ...
Thesis
In this thesis, we prove that the tropical cohomology of a smooth projective tropical variety verifies several symmetry properties: namely, tropical analogs of the Kähler package composed of the Poincaré duality, the hard Lefschetz theorem, the Hodge-Riemann bilinear relations and the monodromy-weight conjecture. We also give some applications.In the local case, we construct a wide family of fans, called tropically shellable fans, whose canonical compactifications verify the Kähler package. We show that the tropical cohomology computes their Chow rings and some quotients of the Stanley-Reisner rings of simplicial complexes which are of particular interest in combinatorics.In the global case, the proof of the main theorem mentioned above uses interesting objects as the existence of some good triangulations and specific versions of tropical analogs of the Deligne spectral sequence, the Steenbrink spectral sequence and the monodromy operator also known as the tropical eigenwave operator.As an application of our results, we get a generalization of the work of De Concini-Procesi and Feichtner-Yuzvinski about wonderful compactifications to the case of toric compactifications induced by unimodular subfans of Bergman fans.In another direction, we prove a tropical Hodge conjecture for smooth projective varieties admitting a rational triangulation: the tropical Hodge classes coincide with the kernel of the monodromy restricted to parts of bidegree (p,p).Finally, we provide a generalization of Symanzik polynomials in higher dimensions. In dimension one, these polynomials appear in combinatorics, in physics and recently in asymptotic Hodge theory. They have many known properties that are still valid in our generalization. This is a first step to understand the asymptotic of some data on degenerating families of complex varieties in any dimension. We also provide a complete description of the exchange graph of independent sets of any matroid.
... The following is a useful result on ranks of restriction and contraction of vector spaces and on relating restriction and contraction through orthogonality [12,7]. ...
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In this paper we show how to compute port behaviour of multiports which have port equations of the general form BvPQiP=s,B v_P - Q i_P = s, which cannot even be put into the hybrid form, indeed may have number of equations ranging from 0 to 2n, where n is the number of ports. We do this through repeatedly solving with different source inputs, a larger network obtained by terminating the multiport by its adjoint through a gyrator. The method works for linear multiports which are consistent for arbitrary internal source values and further have the property that the port conditions uniquely determine internal conditions. We also present the most general version of maximum power transfer theorem possible. This version of the theorem states that `stationarity' (derivative zero condition) of power transfer occurs when the multiport is terminated by its adjoint, provided the resulting network has a solution. If this network does not have a solution there is no port condition for which stationarity holds. This theorem does not require that the multiport has a hybrid immittance matrix.
... We then find a graph G such that M ′ = M(G). Checking whether or nor M ′ is graphic relies on a result of Tutte [8], which gives a method for determining when a binary matroid is graphic. Checking that M = M(G) is then fairly straightforward, all that is required is to check all complete stars of G, and, if these are all cocircuits of M, then M = M(G). ...
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A {\em connectivity function} on a set E is a function λ:2ER\lambda:2^E\rightarrow \mathbb R such that λ()=0\lambda(\emptyset)=0, that λ(X)=λ(EX)\lambda(X)=\lambda(E-X) for all XEX\subseteq E, and that λ(XY)+λ(XY)λ(X)+λ(Y)\lambda(X\cap Y)+\lambda(X\cup Y)\leq \lambda(X)+\lambda(Y) for all X,YEX,Y \subseteq E. Graphs, matroids and, more generally, polymatroids have associated connectivity functions. In this paper we give a method for identifying when a connectivity function comes from a graph. This method uses no more than a polynomial number of evaluations of the connectivity function. In contrast, we show that the problem of identifying when a connectivity function comes from a matroid cannot be solved in polynomial time. We also show that the problem of identifying when a connectivity function is not that of a matroid cannot be solved in polynomial time.
... The reason underlying why greedy algorithms are effective at finding minimum spanning trees is that the set of forests of a graph forms a graphic matroid [90,85]. ...
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Typing methods are widely used in the surveillance of infectious diseases, outbreaks investigation and studies of the natural history of an infection. And their use is becoming standard, in particular with the introduction of High Throughput Sequencing (HTS). On the other hand, the data being generated is massive and many algorithms have been proposed for phylogenetic analysis of typing data, addressing both correctness and scalability issues. Most of the distance-based algorithms for inferring phylogenetic trees follow the closest-pair joining scheme. This is one of the approaches used in hierarchical clustering. And although phylogenetic inference algorithms may seem rather different, the main difference among them resides on how one defines cluster proximity and on which optimization criterion is used. Both cluster proximity and optimization criteria rely often on a model of evolution. In this work we review, and we provide an unified view of these algorithms. This is an important step not only to better understand such algorithms, but also to identify possible computational bottlenecks and improvements, important to deal with large data sets.
... As for simplicial complexes, there are definitions of matroids in which x ⊂ E is required to be non-empty (like[65]) and definitions, where x = ∅ is included. We comment on this at the end. ...
Article
A finite abstract simplicial complex G defines a matrix L, where L(x,y)=1 if two simplicies x,y in G intersect and where L(x,y)=0 if they don't. This matrix is always unimodular so that the inverse g=L−1 has integer entries g(x,y). In analogy to Laplacians on Euclidean spaces, these Green function entries define a potential energy between two simplices x,y. We prove that the total energy E(G)=∑x,yg(x,y) is equal to the Euler characteristic χ(G) of G and that the number of positive minus the number of negative eigenvalues of L is equal to χ(G).
... Regular chain groups are closely related to regular matroids and totally unimodular matrices. We refer, for example, to Camion [5], Oxley [25], Schrijver [27], Tutte [32], [33], Welsh [37] for proofs and more details. ...
Preprint
In this note we give a polynomial time algorithm for solving the closest vector problem in the class of zonotopal lattices. Zonotopal lattices are characterized by the fact that their Voronoi cell is a zonotope, i.e. a projection of a regular cube. Examples of zonotopal lattices include lattices of Voronoi's first kind and tensor products of root lattices of type A. The combinatorial structure of zonotopal lattices can be described by regular matroids/totally unimodular matrices. We observe that a linear algebra version of the minimum mean cycling canceling method can be applied for efficiently solving the closest vector problem in zonotopal lattices.
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The foundation of a matroid is a canonical algebraic invariant which classifies, in a certain precise sense, all representations of the matroid up to rescaling equivalence. Foundations of matroids are pastures , a simultaneous generalization of partial fields and hyperfields which are special cases of both tracts (as defined by the first author and Bowler) and ordered blue fields (as defined by the second author). Using deep results due to Tutte, Dress–Wenzel, and Gelfand–Rybnikov–Stone, we give a presentation for the foundation of a matroid in terms of generators and relations. The generators are certain “cross-ratios” generalizing the cross-ratio of four points on a projective line, and the relations encode dependencies between cross-ratios in certain low-rank configurations arising in projective geometry. Although the presentation of the foundation is valid for all matroids, it is simplest to apply in the case of matroids without large uniform minors . i.e., matroids having no minor corresponding to five points on a line or its dual configuration. For such matroids, we obtain a complete classification of all possible foundations. We then give a number of applications of this classification theorem, for example: We prove the following strengthening of a 1997 theorem of Lee and Scobee: every orientation of a matroid without large uniform minors comes from a dyadic representation, which is unique up to rescaling. For a matroid M M without large uniform minors, we establish the following strengthening of a 2017 theorem of Ardila–Rincón–Williams: if M M is positively oriented then M M is representable over every field with at least 3 elements. Two matroids are said to belong to the same representation class if they are representable over precisely the same pastures. We prove that there are precisely 12 possibilities for the representation class of a matroid without large uniform minors, exactly three of which are not representable over any field.
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Graphical functions are positive functions on the punctured complex plane C{0,1}\mathbb{C}\setminus\{0,1\} which arise in quantum field theory. We generalize a parametric integral representation for graphical functions due to Lam, Lebrun and Nakanishi, which implies the real analyticity of graphical functions. Moreover we prove a formula that relates graphical functions of planar dual graphs.
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The results obtained in this paper grew from an attempt to generalize the main theorem of [1]. There it was shown that any circuit injection (a 1-1 onto edge map f such that if C is a circuit then f(C) is a circuit) from a 3-connected, not necessarily finite graph G onto a graph H is induced by a vertex isomorphism, where H is assumed to not have any isolated vertices. In the present article we examine the situation when the 1-1 condition is dropped (Chapter 1). An interesting result then is that the theorem remains true for finite (3-connected) graphs G but not for infinite G. In Chapter 2 we retain the 1-1 condition but allow the image of f to be first an arbitrary matroid and second a binary matroid. An interesting result then is the following. Let G be a graph of even order. Then the statement "no nontrivial map f:=>M exists, where M is a binary matroid" is equivalent to "G is Hamiltonian". If G is a graph of odd order, then the statement "no nontrivial map f:G=>M exists, where M is a binary matroid" is equivalent to "G is almost Hamiltonian", where we define a graph G of order n to be almost Hamiltonian if every subset of vertices of order n-1 is contained in some circuit of G. [1] J.H. Sanders and D. Sanders, Circuit preserving edge maps, J. Combin. Theory Ser. B 22 (1977),91-96.
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Given a {0,1}\{0,1\}-matrix M, the graph realization problem for M asks if there exists a spanning forest such that the columns of M are incidence vectors of paths in the forest. The problem is closely related to the recognition of network matrices, which are a large subclass of totally unimodular matrices and have many applications in mixed-integer programming. Previously, Bixby and Wagner have designed an efficient algorithm for graph realization that grows a submatrix in a column-wise fashion whilst maintaining a graphic realization. This paper complements their work by providing an algorithm that works in a row-wise fashion and uses similar data structures. The main challenge in designing efficient algorithms for the graph realization problem is ambiguity as there may exist many graphs realizing M. The key insight for designing an efficient row-wise algorithm is that a graphic matrix is uniquely represented by an SPQR tree, a graph decomposition that stores all graphs with the same set of cycles. The developed row-wise algorithm uses data structures that are compatible with the column-wise algorithm and can be combined with the latter to detect maximal graphic submatrices.
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Group formation tends to involve peer effects. In the presence of such complementarities, however, coalitional games need not have a nonempty core. With a restricted preference structure, I provide new sufficient conditions for the nonemptiness of the core of network games that involve pairwise complementarities between peers. The conditions are twofold: (a) sign-consistency—all agents agree on the sign of the value of any link—and (b) sign-balance—the enemy of my enemy is my friend. My conditions provide a game-theoretic explanation for the longevity of the dichotomy of political alliances in the contemporary world.
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Even-cycle matroids are elementary lifts of graphic matroids and even-cut matroids are elementary lifts of cographic matroids. We present a polynomial algorithm to check if a binary matroid is an even-cycle matroid and we present a polynomial algorithm to check if a binary matroid is an even-cut matroid. These two algorithms rely on a polynomial algorithm (to be described in a pair of follow-up papers) to check if a binary matroid is pinch-graphic.
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In this paper, we study algorithmic questions concerning products of matrices and their consequences for recognition algorithms for polyhedra. The 1-product of matrices S1∈Rm1×n1,S2∈Rm2×n2 is a matrix in R(m1+m2)×(n1n2) whose columns are the concatenation of each column of S1 with each column of S2. The k-product generalizes the 1-product, by taking as input two matrices S1,S2 together with k−1 special rows of each of those matrices, and outputting a certain composition of S1,S2. Our first result is a polynomial time algorithm for the following problem: given a matrix S, is S a k-product of some matrices, up to permutation of rows and columns? Our algorithm is based on minimizing a symmetric submodular function that expresses mutual information from information theory. Our study is motivated by a close link between the 1-product of matrices and the Cartesian product of polytopes, and more generally between the k-product of matrices and the glued product of polytopes. These connections rely on the concept of a slack matrix, which gives an algebraic representation of classes of affinely equivalent polytopes. The slack matrix recognition problem is the problem of determining whether a given matrix is a slack matrix. This is an intriguing problem whose complexity is unknown. Our algorithm reduces the problem to instances which cannot be expressed as k-products of smaller matrices. In the second part of the paper, we give a combinatorial interpretation of k-products for two well-known classes of polytopes: 2-level matroid base polytopes and stable set polytopes of perfect graphs. We also show that the slack matrix recognition problem is polynomial-time solvable for such polytopes. Those two classes are special cases of 2-level polytopes, for which we conjecture that the slack matrix recognition problem is polynomial-time solvable.
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We extend the splitting operation from binary matroids (Raghunathan et al., 1998) to p-matroids, where p-matroids refer to matroids representable over GF(p). We also characterize circuits, bases, and independent sets of the resulting matroid. Sufficient conditions to yield Eulerian p-matroids from Eulerian and non-Eulerian p-matroids by applying the splitting operation are obtained. A necessary and sufficient condition for preserving the connectedness of the p-matroids under splitting operation is also provided.
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In this paper, we bound the integrality gap and the approximation ratio for maximum plane multiflow problems and deduce bounds on the flow-multicut-gap. We consider instances where the union of the supply and demand graphs is planar and prove that there exists a multiflow of value at least half the capacity of a minimum multicut. We then show how to convert any multiflow into a half-integer flow of value at least half the original multiflow. Finally, we round any half-integer multiflow into an integer multiflow, losing at most half the value thus providing a 1/4-approximation algorithm and integrality gap for maximum integer multiflows in the plane.
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In [3], Nathan Bowler and the first author introduced a category of algebraic objects called tracts and defined the notion of (weak and strong) matroids over a tract. In the first part of the paper, we summarize and clarify the connections to other algebraic objects which have previously been used in connection with matroid theory. For example, we show that both partial fields and hyperfields are fuzzy rings, that fuzzy rings are tracts, and that these relations are compatible with previously introduced matroid theories. We also show that fuzzy rings are ordered blueprints in the sense of the second author. Thus fuzzy rings lie in the intersection of tracts with ordered blueprints; we call the objects of this intersection idylls. We then turn our attention to constructing moduli spaces for (strong) matroids over idylls. We show that, for any non-empty finite set E, the functor taking an idyll F to the set of isomorphism classes of rank-r strong F-matroids on E is representable by an ordered blue scheme Mat(r,E). We call Mat(r,E) the moduli space of rank-r matroids on E. The construction of Mat(r,E) requires some foundational work in the theory of ordered blue schemes; in particular, we provide an analogue for ordered blue schemes of the “Proj” construction in algebraic geometry, and we show that line bundles and their global sections control maps to projective spaces, much as in the usual theory of schemes. Idylls themselves are field objects in a larger category which we call F1±-algebras; roughly speaking, idylls are to F1±-algebras as hyperfields are to hyperrings. We define matroid bundles over ordered blue F1±-schemes and show that Mat(r,E) represents the functor taking an ordered blue F1±-scheme X to the set of isomorphism classes of rank-r (strong) matroid bundles on E over X. This characterizes Mat(r,E) up to (unique) isomorphism. Finally, we investigate various connections between the space Mat(r,E) and known constructions and results in matroid theory. For example, a classical rank-r matroid M on E corresponds to a morphism Spec(K)→Mat(r,E), where K (the “Krasner hyperfield”) is the final object in the category of idylls. The image of this morphism is a point of Mat(r,E) to which we can canonically attach a residue idyll kM, which we call the universal idyll of M. We show that morphisms from the universal idyll of M to an idyll F are canonically in bijection with strong F-matroid structures on M. Although there is no corresponding moduli space in the weak setting, we also define an analogous idyll kMw which classifies weak F-matroid structures on M. We show that the unit group of kMw can be canonically identified with the Tutte group of M, originally introduced by Dress and Wenzel. We also show that the sub-idyll kMf of kMw generated by “cross-ratios”, which we call the foundation of M, parametrizes rescaling classes of weak F-matroid structures on M, and its unit group coincides with the inner Tutte group of M. As sample applications of these considerations, we show that a matroid M is regular if and only if its foundation is the regular partial field (the initial object in the category of idylls), and a non-regular matroid M is binary if and only if its foundation is the field with two elements. From this, we deduce for example a new proof of the fact that a matroid is regular if and only if it is both binary and orientable.
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We introduce an object called a tree growing sequence (TGS) in an effort to generalize bijective correspondences between G-parking functions, spanning trees, and the set of monomials in the Tutte polynomial of a graph G. A tree growing sequence determines an algorithm which can be applied to a single function, or to the set PG,q of G-parking functions. When the latter is chosen, the algorithm uses splitting operations – inspired by the recursive definition of the Tutte polynomial – to iteratively break PG,q into disjoint subsets. This results in bijective maps τ and ρ from PG,q to the spanning trees of G and Tutte monomials, respectively. We compare the TGS algorithm to Dhar’s algorithm and the family described by Chebikin and Pylyavskyy in 2005. Finally, we compute a Tutte polynomial of a zonotopal tiling using analogous splitting operations.
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TÜBA-EMAN symposium, titled as “Exploring the Commonalities of the Mediter-ranean Region”, was held two years ago. It was our wish to publish the proceedings in the aftermath of the symposium. The publication has been delayed due to, among other reasons, the transitional period that TUBA has gone since then. I am glad that the papers, except three of them, have now been compiled and TUBA is pleased to publish the Proceedings. The Proceedings comprises the full texts of the papers, the original symposium programme and abstracts of the presented papers. TUBA sees itself as part of the global scientific community and values its coopera-tion with sister academies and inter-academy organizations. We shall do our best to contribute towards the activities of EMAN in the future as well. I would like to express my thanks to those academicians who showed relentless efforts in organizing this symposium, most notably Pavao Rudan, the President of EMAN, Ahmet Cevat Acar, the then President of TUBA, and A. Nuri Yurdusev, the then Vice President of TUBA. I would also like to thank the contributors to this Pro-ceedings and our staff in TUBA as well.
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