Products of abstract polytopes

Abstract

Given two convex polytopes, the join, the cartesian product and the direct sum of them are well understood. In this paper we extend these three kinds of products to abstract polytopes and introduce a new product, called the topological product, which also arises in a natural way. We show that these products have unique prime factorization theorems. We use this to compute the automorphism group of a product in terms of the automorphism groups of the factors and show that (non trivial) products are almost never regular or two-orbit polytopes. We finish the paper by studying the monodromy group of a product, show that such a group is always an extension of a symmetric group, and give some examples in which this extension splits. arXiv:1603.03585v1 [math.CO]• If P is the prism (or the bipyramid) over a 3-polytope having the property that all its vertex figures (faces) are isomorphic to an n-gon, with n not congruent to 0 modulo 9;• If = and each Q i has rank 2.By [2], we already know that the monodromy group of a pyramid over an n-gon is an extension of S 4 by (C m ) 4 , where m = p gcd (3,p) , and that such extension splits whenever n is not congruent to 0 modulo 9. Our techniques to show Theorem B can also be used to show this result. Abstract polytopesAbstract polytopes generalise the (face-lattice) of classical polytopes as combinatorial structures. In this chapter we review the basic theory of abstract polytopes and refer the reader to [11] for a detail exposition of the subject.An (abstract) n-polytope (or (abstract) polytope of rank n) P is a partially ordered set whose elements are called faces that satisfies the following properties. It contains a minimum face F −1 and a maximum

Voltage operations on maniplexes

“…Both the prism and the pyramid over an n-polytope are (n + 1)-polytopes. They are particular cases of products of polytopes, in the sense of [9]: the pyramid is the join product by a vertex, while the prism is the direct product by an edge.…”

Categories: Between Cubes and Globes. Sketch I

“…Note that, for an element ∈ of a lower set, ∖ { } is a lower set, iff ∈ max . So, we obtain a collection of cones indexed by pairs ( , ) of a lower subset ⊆ and a function : max → 2 := {0, 1}, where According to [4], one can define four types of products (join, sum, direct product, topological product) for a family ( ) ∈ of abstract polytopes taking a Cartesian product of undelying posets for , > , < , and <> , respectively. The construction − turns the coproduct into a direct product and the ordered sum to an ordered sum: Proposition 3.…”

The antiprism of an abstract polytope

“…The paper uses some of the ideas and notation of the products of polytopes described in [8]. Moreover, we give relations between some products and their antiprisms.…”

“…In this section we give the basic definitions from the theory of abstract polytopes, as well as two of their products. For details on these subjects we refer the reader to [15] and [8], respectively. An (abstract) polytope is a partially ordered set (poset) P, whose elements are called faces, such that it has a minimal and a maximal element and is ranked: all its maximal chains, called flags, have the same number of elements.…”

“…In [8] the authors study the automorphism group of a product in terms of the automorphisms groups of the factors. In particular we have the following result.…”