Hypergraph polytopes

Došen, Kosta; Petric, Z.

doi:10.1016/j.topol.2011.05.015

Cited by 22 publications

(59 citation statements)

References 25 publications

(104 reference statements)

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“…With these two kinds of arrows mixed we obtain also in three dimensions another kind of polyhedron, which was called hemiassociahedron in [DP10] (Example 5.14). That paper investigates in particular the relationship of the hemiassociahedron to the permutohedron (from which, together with [Ton97], one may gather that the hemiassociahedron, conceived as an abstract polytope, can be realized; we will however not go here into this problem, for which one should also consult [DP11]). …”

Section: Womentioning

confidence: 99%

“…In Section 13 we consider how the mixed coherence conditions engender a new kind of polyhedron, related to, but different from, the three-dimensional associahedron and permutohedron. The family to which this polyhedron, called hemiassociahedron, belongs (the family includes also the three-dimensional cyclohedron) is investigated in [DP10] and [DP11] (which should be consulted for further references). The mixed conditions may also however lead in some cases to the three-dimensional associahedron and permutohedron, which provides another perspective on these well-known polyhedra tied to associativity and commutativity.…”

Section: Introductionmentioning

confidence: 99%

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Weak Cat-Operads

Došen¹,

Petric²

2015

Logical Methods in Computer Science

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Abstract. An operad (this paper deals with non-symmetric operads) may be conceived as a partial algebra with a family of insertion operations, which correspond to substitution of an operation within an operation. These insertion operations are Gerstenhaber's circle-i products, and they satisfy two kinds of associativity, one of them involving commutativity. A Cat-operad is an operad enriched over the category Cat of small categories, as a 2-category with small hom-categories is a category enriched over Cat. This means that the operadic operations of the same arity in a Cat-operad do not make just a set, but they are the objects of a small category. The notion of weak Cat-operad is to the notion of Cat-operad what the notion of bicategory is to the notion of 2-category. This means that the equations of operads like associativity of insertions are replaced by isomorphisms in a category. The goal of this paper is to formulate conditions concerning these isomorphisms that ensure coherence, in the sense that all diagrams of canonical arrows commute. This is the sense in which the notions of monoidal category and bicategory are coherent. (The coherence of monoidal categories, which is due to Mac Lane, is the best known coherence result.) The coherence proof in the paper is much simplified by indexing the insertion operations in a context-independent way, and not in the usual manner. This proof, which is in the style of term rewriting, involves an argument with normal forms that generalizes what is established with the completeness proof for the standard presentation of symmetric groups. This generalization may be of an independent interest, and related to other matters than those studied in this paper. Some of the coherence conditions for weak Cat-operads lead to the hemiassociahedron, which is a polyhedron related to, but different from, the three-dimensional associahedron and permutohedron.

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Section: Womentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Weak Cat-Operads

Došen¹,

Petric²

2015

Logical Methods in Computer Science

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“…This is an alternative, inductive approach to the same matters, which is given in [10]. For α a finite set, let H ⊆ P (α) be such that ∅ ∈ H and α = H. Then H is a hypergraph on α (see [1], Section 1.1).…”

Section: Abstract Polytopes Of Hypergraphsmentioning

confidence: 99%

“…For α a finite set, let H ⊆ P (α) be such that ∅ ∈ H and α = H. Then H is a hypergraph on α (see [1], Section 1.1). A hypergraph H is atomic when for every x ∈ H we have that {x} ∈ H (see [10], Section 3).…”

Section: Abstract Polytopes Of Hypergraphsmentioning

confidence: 99%

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On stretching the interval simplex-permutohedron

Petric

2013

J Algebr Comb

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A family of polytopes introduced by E.M. Feichtner, A. Postnikov and B. Sturmfels, which were named nestohedra, consists in each dimension of an interval of polytopes starting with a simplex and ending with a permutohedron. This paper investigates a problem of changing and extending the boundaries of these intervals. An iterative application of Feichtner-Kozlov procedure of forming complexes of nested sets is a solution of this problem. By using a simple algebraic presentation of members of nested sets it is possible to avoid the problem of increasing the complexity of the structure of nested curly braces in elements of the produced simplicial complexes.Mathematics Subject Classification (2010): 05C65, 55U10, 52B11, 52B12, 51M20

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Celebrating Loday’s associahedron

Pilaud,

Santos,

Ziegler

2023

Arch. Math.

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We survey Jean-Louis Loday’s vertex description of the associahedron, and its far reaching influence in combinatorics, discrete geometry, and algebra. We present in particular four topics where it plays a central role: lattice congruences of the weak order and their quotientopes, cluster algebras and their generalized associahedra, nested complexes and their nestohedra, and operads and the associahedron diagonal.

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Hypergraph polytopes

Cited by 22 publications

References 25 publications

Weak Cat-Operads

Weak Cat-Operads

On stretching the interval simplex-permutohedron

Celebrating Loday’s associahedron

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