# Chain

A **chain** is a set of elements in a polytope that are all incident to each other. Succinctly, it is a subset of a flag. For example, in a polyhedron, some types of chains include "edge-faces" (a face and an edge that touch each other), "vertex-faces," "vertex-edges," flags (comprising a vertex, edge, and face), and individual elements. It follows from the definition of an abstract polytope that all elements in a chain must have different rank; e.g. no two vertices can be incident.

Chains are more generally defined for incidence geometries, which impose that all elements in a chain must have different type.

## Definition[edit | edit source]

### Incidence geometry[edit | edit source]

In their most abstract, the definition of a chain only relies on a concept of incidence, and so chains can be defined for any incidence system.

A chain is a set of elements C such that any two elements of C are incident with each other.

Confusingly in the context of incidence geometries chains are called "flags", although that name has a different meaning in most other contexts.

### Incidence graph[edit | edit source]

A chain is a clique in the incidence graph of a structure.

### Poset[edit | edit source]

Most commonly chains are defined in terms of partially ordered sets, or "posets". In this context a chain of a poset 𝓚 is a total suborder. That is it is a subset of the elements of 𝓚 such that no pair of elements in the subset are incomparable.

This definition applies to cases that the definition in terms of incidence geometry does not since a poset does not necessarily need to be ranked. However it does not apply to structures, such as hypertopes, which have no inherent ranking to their elements, whereas the definition in terms of incidence geometry can.

## Types of chains[edit | edit source]

### Flags[edit | edit source]

Flags are a particular kind of chain. They can be defined two ways:

- As a maximal chain, that is a chain that is not a strict subset of any other chain.
- As a chain that contains an element of every type (rank). These are also called
**chambers**.

In most structures these definitions are equivalent, in particular they are equivalent in any incidence system.

### Darts[edit | edit source]

A **dart** or a **half-edge**^{[note 1]} is a type of polyhedral chain. In a polyhedron 𝓟, a dart is a chain consisting of a vertex and an edge.

Darts are used in rotation systems, similar to how flags are used graph-encoded maps.

A dart-transitive map is called a regular map although it is not necessarily regular in the ordinary sense.

### Blade[edit | edit source]

A **blade** is a chain consisting of a face, an edge, and a vertex. For polyhedra blades are the same as flags.

## Flag vectors[edit | edit source]

The **flag vector** of a polytope comprises the number of chains of each type. For example, the flag vector of a polyhedron contains the number of vertices, edges, faces, vertex-edge chains, face-edge chains, face-vertex chains, and face-edge-vertex chains. (Conventionally, the improper elements are left out as they don't add any new information.)

## Notes[edit | edit source]

- ↑ Some authors use the term "half-edge" to refer to any pair of adjacent flags in a polyhedron.

## External links[edit | edit source]

- Wedd, N. Glossary. Gives definitions for several chain-related terms.