Abstract
We consider non-empty sets endowed with an order relation denoted by ⊂, read “is a face of” or “is contained in”. Such a set is called a complex if the ordered subset of all faces of any given element is isomorphic with the ordered set of all subsets of a set, and if any two elements A,B have a greatest lower bound, denoted by A ∩ B. A complex has a smallest element which we shall always denote by O. The number of minimal non O faces of an element A is called the rank of A, and denoted by rk A. The elements of rank 1 are called vertices. Since an element of a complex is completely characterized by the set of its vertices, we may also define a complex as a set Δ of subsets of a set V (the set of vertices), such that {x}∈ Δ for all x ∈ V, and that B ⊂ A ∈ Δ implies B ∈ Δ; the rank of an element of Δ is its cardinality (as a subset of V ). The rank of a complex Δ, denoted by rk Δ, is by definition sup {rk A ∣ A ∈ Δ}. A complex is called a simplex if it is isomorphic to the set of all subsets of a given set, ordered by inclusion. Hereafter, Δ always denotes a complex.
Keywords
- Order Relation
- Maximal Element
- Involutory Automorphism
- Small Convex
- Adjacent Chamber
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
This is a preview of subscription content, access via your institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
Rights and permissions
Copyright information
© 1974 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
(1974). Complexes. In: Buildings of Spherical Type and Finite BN-Pairs. Lecture Notes in Mathematics, vol 386. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-38349-9_1
Download citation
DOI: https://doi.org/10.1007/978-3-540-38349-9_1
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-06757-3
Online ISBN: 978-3-540-38349-9
eBook Packages: Springer Book Archive
