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We study the combinatorics of tropical hyperplane arrangements, and their relationship to (classical) hyperplane face monoids. We show that the refinement operation on the faces of a tropical hyperplane arrangement, introduced by Ardila and Develin in their definition of a tropical oriented matroid, induces an action of the hyperplane face monoid of the classical braid arrangement on the arrangement, and hence on a number of interesting related structures. Along the way, we introduce a new characterization of the types (in the sense of Develin and Sturmfels) of points with respect to a tropical hyperplane arrangement, in terms of partial bijections which attain permanents of submatrices of a matrix which naturally encodes the arrangement.
We study the combinatorics of hyperplane arrangements over arbitrary fields. Specifically, we determine in which situation an arrangement and its reduction modulo a prime number have isomorphic lattices via the use of minimal strong $sigma$-Grobner b
There is a trinity relationship between hyperplane arrangements, matroids and convex polytopes. We expand it as resolving the complexity issue expected by Mnevs universality theorem and conduct combinatorializing so the theory over fields becomes rea
In this article, we study the weak and strong Lefschetz properties, and the related notion of almost revlex ideal, in the non-Artinian case, proving that several results known in the Artinian case hold also in this more general setting. We then apply
We show that the deletion theorem of a free arrangement is combinatorial, i.e., whether we can delete a hyperplane from a free arrangement keeping freeness depends only on the intersection lattice. In fact, we give an explicit sufficient and necessar
We introduce a new algebra associated with a hyperplane arrangement $mathcal{A}$, called the Solomon-Terao algebra $mbox{ST}(mathcal{A},eta)$, where $eta$ is a homogeneous polynomial. It is shown by Solomon and Terao that $mbox{ST}(mathcal{A},eta)$ i