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In mathematics, an abstract polytope is an algebraic partially ordered set (poset) which captures the combinatorial properties of a traditional polytope (a generalisation of polygons and polyhedra into any number of dimensions) without specifying purely geometric properties such as angles or edge lengths.

An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope. The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.

Formal definition

Preliminary concepts

Let $\{P,\leq \}$ be a partially ordered set. Then we make the following definitions: Incidence: We say that faces $F,G\in P$ are incident if $F\leq G$ or $G\leq F$. Chain: A [chain] in $P$ is a subset of $P$ which is [totally ordered] by the order on $P$. Flag: A flag is a maximal chain; that is it is a chain which is not contained as a strict subset of some larger chain. Section: For any $F,G\in P$, a set of the form $\{H\in P:F\leq H\leq G\}$ is called a section of $P$ and this set is denoted by $G/F$ Rank: The rank of a face $F$ is the length of the any chain in the section $F/F_{-1}$ where $F_{-1}$ is a minimal element of $P$.

Note that the conditions (P1), (P2) below descend to any subset of $P$, so the rank of a face in an abstract polytope is indeed well-defined

Definition

An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:

1. It has exactly one [least face] and exactly one [greatest face].
2. All flags contain the same number of faces.
3. It is strongly connected.
4. If the ranks of two faces a > b differ by 2, then there are exactly 2 faces that lie strictly between a and b.

An n-polytope is a polytope of rank n. The abstract polytope associated with a real convex polytope is also referred to as its face lattice.

Introductory concepts

Traditional versus abstract polytopes

Six geometric quadrilaterals.

In Euclidean geometry, the six quadrilaterals illustrated are all different. Yet they have a common structure in the alternating chain of four vertices and four sides which gives them their name. They are said to be isomorphic or “structure preserving”.

This common structure may be represented in an underlying abstract polytope, a purely algebraic partially-ordered set which captures the pattern of connections or incidences between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.

What is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa. For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope.

The Hasse diagram

The graph (left) and Hasse diagram of a quadrilateral, showing ranks (right)

Smaller posets, and polytopes in particular, are often best visualised in a Hasse diagram, as shown. By convention, faces of equal rank are placed on the same vertical level. Each "line" between faces, say F, G, indicates an ordering relation < such that F < G where F is below G in the diagram.

The Hasse diagram defines the unique poset and therefore fully captures the structure of the polytope. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice versa. The same is not generally true for the graph representation of polytopes.

Faces and ordering

In an abstract polytope, each structural element - vertex, edge, cell, etc. is associated with a corresponding member of the set. The term face here usually refers to any such element e.g. a vertex (0-face), edge (1-face) or a general k-face, and not just a polygonal 2-face.

The faces are given an "ordering", where we consider a face F to be "less than" a face G, written F≤G, whenever F is to be thought of as a "subface" of G (including the case where F=G), for example when F is a vertex of an edge G. We say that F and G are incident if F≤G or G≤F. This usage of "incidence" also occurs in finite geometry, although it differs from traditional geometry and some other areas of mathematics. For example in the square abcd, edges ab and bc are not incident in the sense of abstract polytopes, although they are incident in the traditional sense because they share the vertex b.

Rank

There is also a number which we can associate to each face, which we call the rank of the face. Traditionally this corresponds to the dimension of each face. F vertices have rank = 0, edges rank = 1 and so on. The rank of a face F is formally defined as (m − 2), where m is the maximum number of faces in any chain (F', F", ... , F) satisfying F' < F" < ... < F. F' is always the least face, F₋₁.

The rank of an abstract polytope P is the maximum rank n of any face. It is always the rank of the greatest face F_n.

For some ranks, their face-types are named in the following table.

Rank	-1	0	1	2	3	...	n - 2	n - 1	n
Face Type	Least	Vertex	Edge	†	Cell		Subfacet or ridge	Facet	Greatest

† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of any rank.

Least and greatest faces

Each abstract polytope has a maximal and a minimal face with respect to the partial order ≤, which are often called the least face and the greatest face respectively. Since the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as F₋₁. It is conventional to identify the greatest face with the entire polytope and to identify the least face with the empty set ∅. It is not usually realized. This is called the greatest face. In an n-dimensional polytope, the greatest face has rank = n and may be denoted as F_n. It is sometimes realized as the interior of the geometric figure.

These least and greatest faces are sometimes called improper faces, with all others being proper faces.

Sections

The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (red), and a 2-section (green).

Any subset P' of a poset P is a poset (with the same relation <, restricted to P').

In an abstract polytope, given any two faces F, H of P with F ≤ H, the set {G | F ≤ G ≤ H} is called a section of P, and denoted H/F. (In order theory, a section is called a closed interval of the poset and denoted [F, H].

For example, in the prism abcxyz (see diagram) the section xyz/ø (highlighted green) is the triangle

{ø, x, y, z, xy, xz, yz, xyz}.

A k-section is a section of rank k.

P is thus a section of itself.

This concept of section does not have the same meaning as in traditional geometry.

Connectedness

A poset P is connected if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces

H₁, H₂, ... ,H_k

such that F = H₁, G = H_k, and each H_i, i < k, is incident with its successor.

The above condition ensures that a pair of disjoint triangles abc and xyz is not a (single) polytope.

A poset P is strongly connected if every section of P (including P itself) is connected.

With this additional requirement, two pyramids that share just a vertex are also excluded. However, two square pyramids, for example, can, be "glued" at their square faces - giving an octahedron. The "common face" is not then a face of the octahedron.

Flags

A flag is a maximal chain of faces, i.e. a (totally) ordered set Ψ of faces, each a subface of the next (if any), and such that Ψ is not a subset of any larger chain. Given any two distinct faces F, G in a flag, either F < G or F > G.

For example, {ø, a, ab, abc} is a flag in the triangle abc.

For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.

Facets

The facet for a given j-face F is the (j−1)-section F/∅, where F_j is the greatest face.

For example, in the triangle abc, the facet at ab is ab/b = {∅, a, b, ab}, which is a line segment.

The distinction between F and F/∅ is not usually significant and the two are often treated as identical.

Vertex figures

The vertex figure at a given vertex V is the (n−1)-section F_n/V, where F_n is the greatest face.

For example, in the triangle abc, the vertex figure at b is abc/b = {b, ab, bc, abc}, which is a line segment. For example, the vertex figures of a cube are triangles.

A simple example

The faces of the abstract quadrilateral or square are shown in the table below:

Face type	Rank (k)	Count	k-faces
Least	−1	1	F−1
Vertex	0	4	a, b, c, d
Edge	1	4	W, X, Y, Z
Greatest	2	1	G

The relation < comprises a set of pairs, which here include

F₋₁<a, ... , F₋₁<X, ... , F₋₁<G, ... , b<Y, ... , c<G, ... , Z<G.

Order relations are transitive, i.e. F < G and G < H implies that F < H. Therefore, to specify the hierarchy of faces, it is not necessary to give every case of F < H, only the pairs where one is the successor of the other, i.e. where F < H and no G satisfies F < G < H.

The edges W, X, Y and Z are sometimes written as ab, ad, bc, and cd respectively, but such notation is not always appropriate.

All four edges are structurally similar and the same is true of the vertices. The figure therefore has the symmetries of a square and is usually referred to as the square.

Realisations

A traditional geometric polytope is said to be a realisation of the associated abstract polytope. A realisation is a mapping or injection of the abstract object into a real space, typically Euclidean, to construct a traditional polytope as a real geometric figure.

The six quadrilaterals shown are all distinct realisations of the abstract quadrilateral, each with different geometric properties. Some of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realisations. A conventional polytope is a faithful realisation.