By John Rhodes, Pedro V. Silva
This self-contained monograph explores a brand new conception based round boolean representations of simplicial complexes resulting in a brand new type of complexes that includes matroids as relevant to the idea. The e-book illustrates those new instruments to review the classical thought of matroids in addition to their vital geometric connections. additionally, many geometric and topological positive aspects of the idea of matroids locate their opposite numbers during this prolonged context.
Graduate scholars and researchers operating within the parts of combinatorics, geometry, topology, algebra and lattice concept will locate this monograph beautiful a result of wide selection of recent difficulties raised by means of the idea. Combinatorialists will locate this extension of the idea of matroids priceless because it opens new traces of analysis inside of and past matroids. The geometric beneficial properties and geometric/topological purposes will entice geometers. Topologists who wish to practice algebraic topology computations will have fun with the algorithmic strength of boolean representable complexes.
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Extra info for Boolean Representations of Simplicial Complexes and Matroids
Note also that F is a ^-semilattice and therefore a lattice with the determined join. 3. V; H / be a boolean representable simple simplicial complex and let F 2 FSub^ Fl H. ;/: Proof. (i) ) (ii). L; V / 2 LR H. 2. (ii) ) (i). L; V / 2 FLg. 2. Therefore F 2 Im Â. 6) Indeed, every F 2 FSub^ Fl H, being a \-subsemilattice of Fl H, constitutes a lattice of its own right with the determined join. F; V / produces enough witnesses to recognize all the faces of H. 6) holds. Let Lat H denote the Rees quotient (see Sect.
Now let Y 2 Fl H be arbitrary. Suppose that p 2 X n Y . Since ; 2 H \ 2Y , we must have ; [ fpg 2 H , contradicting p … V Ã . Fl H/. (ii) If H is trim, then V Ã D V . 2(iv) that V Â Fl H. Fl H; V / 2 FLg. Indeed, for every X 2 Fl H, we have X D _ffxg j x 2 X g. Fl H; V / defined in Sect. 3 as the boolean representation of a _-generated lattice. 5 that Mat H constitutes somehow the canonical unique biggest boolean representation for H (when H is boolean representable). However, there exist many smaller boolean representations, even in the matroid case.
Our next goal is to build an isomorphism from Â. A first obstacle is the fact that Â is not onto: not every F 2 FSub^ Fl H is rich enough to represent H. Before characterizing the image of Â, it is convenient to characterize which lattices provide boolean representations. 2. L; V / 2 FLg. ;/: Proof. (i) ) (ii). 2. (ii) ) (i). Let X Â V . L; V /. Assume that X 2 H . L; V / and so X is c-independent. 5 The Lattice of Lattice Representations 53 Conversely, assume that X is c-independent. Fl H; V /.