By John Rhodes, Pedro V. Silva
This self-contained monograph explores a brand new thought established round boolean representations of simplicial complexes resulting in a brand new type of complexes that includes matroids as crucial to the speculation. The booklet illustrates those new instruments to review the classical idea of matroids in addition to their vital geometric connections. additionally, many geometric and topological good points of the speculation of matroids locate their opposite numbers during this prolonged context.
Graduate scholars and researchers operating within the components of combinatorics, geometry, topology, algebra and lattice conception will locate this monograph attractive a result of wide selection of recent difficulties raised through the speculation. Combinatorialists will locate this extension of the speculation of matroids priceless because it opens new strains of study inside of and past matroids. The geometric beneficial properties and geometric/topological purposes will attract geometers. Topologists who wish to practice algebraic topology computations will delight in the algorithmic strength of boolean representable complexes.
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Additional resources for Boolean Representations of Simplicial Complexes and Matroids
Let p 2 V n Y and let m I 2 H \2Y . Since X Â X ı Â X ı 2 Â : : : and I is finite, we have I 2 H \2Xı for some m 1. Suppose that I [ fpg … H . Since p 2 V n X ı m , then p 2 X ı mC1 , contradicting p 2 V n Y . Thus I [ fpg 2 H and so Y is closed. Therefore Cl X D Y . Z/ denote the set of all faces of H contained in Z with maximum possible dimension. 2) Indeed, let p 2 X ı n X . Then I [ fpg … H for some I 2 H \ 2X . Since we may replace I by any I 0 2 H \ 2X containing I , We may assume that I is maximal with respect to I 2 H \ 2X and I [ fpg … H .
M / Â Fl M . Fl M; V /. Then M 0 can be obtained from M by successively inserting a row of zeroes (corresponding to the fact that V 2 Fl M ) and sums of rows of M in BjV j (corresponding to intersections of the Zi ). 4(ii). 5 The Lattice of Lattice Representations We show in this section that we can organize the lattice representations of a simple simplicial complex into a lattice of their own. L0 ; A/ if there exists some _-map ' W L ! L0 such that 'jA D 1A . 1. L0 ; A/ 2 FLg. L; A/. Proof. (i) ) (ii).
1) holds and so M is a reduced boolean representation of H as claimed. 7. 4(i) immediately implies that if H admits a reduced boolean representation, then there exists a unique maximal one (up to permuting rows). The main theorem of this section provides a more concrete characterization. 5. V; H / be a simplicial complex. Then the following conditions are equivalent: (i) H has a boolean representation; (ii) Mat H is a reduced boolean representation of H. Moreover, in this case any other reduced boolean representation of H is congruent to a submatrix of Mat H.
Boolean Representations of Simplicial Complexes and Matroids by John Rhodes, Pedro V. Silva