A generalized permutahedron

Pouzet, Maurice; Reuter, Klaus; Rival, Ivan; Zaguia, Nejib

doi:10.1007/bf01181874

Cited by 11 publications

(15 citation statements)

References 11 publications

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“…We start by listing critical pairs of P 6 and P 10 . Observe that crit(P 6 ) = (0, 5), (1,6), (2,4), (3,9), (4, 7), (5, 7), (6,8) crit(P 10 ) = (0, 5), (1, 4), (2, 3), (3,9), (5, 7), (6,8) Observe also that any upper cover of P 6 has no irreducible elements. Therefore it must be one of the ordered sets listed in Fig.…”

Section: Proof Of Theoremmentioning

confidence: 94%

“…The critical pairs of P 2 are (0, 6) and (1, 5), which play symmetric roles (there is an automorphism mapping one to the other), (2, 7), and (3,9), (4,8), which can be mapped with a dual automorphism to the first two critical pairs. Hence we are done if we can show that P 2 ∨ {(0, 6)} and P 2 ∨ {(2, 7)} do not have the fixed point property.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…Our investigation lies within the more general scope of the study of the behavior of properties of orders with respect to the addition of comparabilities (see [1,8,14]). …”

Section: Introductionmentioning

confidence: 99%

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Order Extensions and the Fixed Point Property

Zaguia

2008

Order

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The purpose of this paper is to investigate how the fixed point property and its negation behave when a covering relation is added to the order. We prove that every finite ordered set which is not totally ordered and which is dismantlable by retractables, respectively by irreducibles, has an upper cover (in its extension lattice) which is also dismantlable by retractables, respectively by irreducibles. We also provide examples of finite ordered sets having the fixed point property so that none of their upper covers has the fixed point property.

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Section: Proof Of Theoremmentioning

confidence: 94%

Section: Proof Of Theoremmentioning

confidence: 99%

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Order Extensions and the Fixed Point Property

Zaguia

2008

Order

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“…Nous prouvons essentiellement que les ordres critiques de hauteur au plus trois sont de dimension 2 ; un resultat qui est en rapport avec une conjecture [8,9] sur les ordres perpendiculaires. Notre étude s'inscrit dans le cadre plus général de l'étude du comportement de propriétés d'ordres vis-à-vis de l'adjonction de comparabilités (pour un autre exemple, voir [1,6,7]). …”

Section: Introductionunclassified

Ordres indécomposables critiques

Zaguia

2008

Comptes Rendus. Mathématique

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“…The structure of this set of extensions has been studied from different points of view in [9] and [10]. A classical result in order theory states that the set of all linear extensions of a partial order is in a one-to-one correspondance with the maximal chains of the 626 lattice of ideals of the order.…”

Section: Introductionmentioning

confidence: 99%

Extending a Partially Ordered Set: Links with its Lattice of Ideals

Baldy

Morvan

Thierry

2000

Formal Power Series and Algebraic Combinatorics

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Abstract. A well-known result of Bonnet and Pouzet [3] bijectively links the set of linear extensions of a partial order P with the set of maximal chains of its lattice of ideals I(P). We extend this result by showing that there is a one-to-one correspondence between the set of all extensions of P and the set of all sublattices of I(P) which are chainmaximal in the sense that every chain which is maximal (for inclusion) in the sublattice is also maximal in the lattice. We prove that the absence of an order 8 as a convex suborder of P is equivalent to the absence of 1(8) as a convex suborder of I(P). Let S be a set of partial orders and let us call S-convex-free any order that does not contain any order of S as convex suborder. We deduce from the previous results that there is a one-to-one correspondence between the set of S-convex-free extensions of P and the set of I(S)-convex-free chain-maximal sublattices of I(P). This can be applied to some classical classes of orders (total orders and in the finite case, weak orders, interval orders, N-free orders). In the particular case of total orders this gives as a corollary the result of Bonnet and Pouzet.

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A generalized permutahedron

Cited by 11 publications

References 11 publications

Order Extensions and the Fixed Point Property

Order Extensions and the Fixed Point Property

Ordres indécomposables critiques

Extending a Partially Ordered Set: Links with its Lattice of Ideals

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