Pattern Recognition on Oriented Matroids

2Boolean Intervals

Let V be a finite set, and 2V the simplex {F: F ⊆ V}, that is, the power set of the set V. Recall that a family A ⊆ 2V is called an abstract simplicial complex (or a complex for short) on the vertex set V if, given subsets A and B of the set V, the inclusions A ⊆ B ∈ △ imply A ∈ △, and if {v} ∈ A, for any v − V . If Γ is a complex such that Γ ∈ △ (that is, Γ is a subcomplex of △), then the family △ — ∈ is called a relative simplicial complex.

If Ψ is a relative complex, then the sets F ∈ Ψ are called the faces of Ψ. The dimension dim(F) of a face F is by convention equal to |F| − 1; the cardinality |F| is also called the size of F. If we let # denote the number of sets in a family, and if #Ψ > 0, then the size d(Ψ) of Ψ is ...