, If S is a system of signed sets we denote by , X k ? S} the set of all (finite) compositions of S. The empty set is considered as the empty composition of signed sets, and so ? ? L(S) One can endow L(S) with a partial order relation where Y ? X if and only if Adding a global maximum?1maximum?maximum?1 this partial order becomes a lattice denoted by F big (S) := (L(S) ? { ? 1}, ?) The lattice property is easily seen: the empty set is the global minimum and the unique join of two signed sets X, Y is?1is?is?1 if S(X, Y ) = ? and X ? Y otherwise. There are two important undirected graphs associated to F big (S) ? one on its atoms and one on its coatoms. So the first is a graph G(S) with vertex set S such that two signed sets X, Y ? S are connected by an edge if and only if there is Z ? L(S) ? { ? 1} such that X, Y are the only elements of S with X, Y ? Z. The second graph is called called the tope graph G(T ) It is defined on the set T of coatoms of F big (S) The elements of T are called topes. Topes S, T ? T are contained in an edge of G(T ) if and only if there is Z ? L(S) such that S, T are the only elements of T with X, Y ? Z. Let G be any graph on a system R of signed sets with ground set E. For X 1 , . . . , X k ? R we denote by [X 1 , . . . , X k ] the subgraph of G induced by (e)} for all e ? E}. We call [X 1 , . . . , X k ] the crabbed hull of X 1 , . . . , X k . An (X, Y )-path in G is called crabbed if it is contained in [X, Y ]. If S is the system of cocircuits C * of an oriented matroid M, then the elements of L(C * ) are called the covectors of M. Moreover, F big (C * ) is a graded lattice with rank function r, see [4]. In this case F big (C * ) is called the big face lattice of M. The rank rk(M) of M is defined as r( ? 1) ? 1, i.e., one less than the rank of F big (C * ). Moreover, G(C * ) is called the cocircuit graph of M. One important oriented matroid operation is the contraction, again proofs for its properties can be, ) is {X\A | X ? L(C * ) and A ? X 0 }. Furthermore, for U ? L(C * ) we have rk(M/U 0 ) = r(U ), where r(U ) is the rank of U in F big (C * )
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