scholarly journals The numbers of edges of the order polytope and the chain polytope of a finite partially ordered set

2017 ◽  
Vol 340 (5) ◽  
pp. 991-994 ◽  
Author(s):  
Takayuki Hibi ◽  
Nan Li ◽  
Yoshimi Sahara ◽  
Akihiro Shikama
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Order Polytope ◽  
Finite Partially Ordered Set

scholarly journals Unimodular Equivalence of Order and Chain Polytopes

2016 ◽  
Vol 118 (1) ◽  
pp. 5 ◽  
Author(s):  
Takayuki Hibi ◽  
Nan Li
Keyword(s):  
Commutative Algebra ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered ◽  
Order Polytope ◽  
Finite Partially Ordered Set

Order polytope and chain polytope are two polytopes that arise naturally from a finite partially ordered set. These polytopes have been deeply studied from viewpoints of both combinatorics and commutative algebra. Even though these polytopes possess remarkable combinatorial and algebraic resemblance, they seem to be rarely unimodularly equivalent. In the present paper, we prove the following simple and elegant result: the order polytope and chain polytope for a poset are unimodularly equivallent if and only if that poset avoid the 5-element "X" shape subposet. We also explore a few equivalent statements of the main result.

scholarly journals Faces of 2-Dimensional Simplex of Order and Chain Polytopes

Mathematics ◽  
2019 ◽  
Vol 7 (9) ◽  
pp. 851
Author(s):  
Aki Mori
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Explicit Description ◽  
Dimensional Simplex ◽  
Partially Ordered ◽  
The Face ◽  
Finite Partially Ordered Set ◽  
Arbitrary Triangle

Each of the descriptions of vertices, edges, and facets of the order and chain polytope of a finite partially ordered set are well known. In this paper, we give an explicit description of faces of 2-dimensional simplex in terms of vertices. Namely, it will be proved that an arbitrary triangle in 1-skeleton of the order or chain polytope forms the face of 2-dimensional simplex of each polytope. These results mean a generalization in the case of 2-faces of the characterization known in the case of edges.

Planar Lattices

1975 ◽  
Vol 27 (3) ◽  
pp. 636-665 ◽  
Author(s):  
David Kelly ◽  
Ivan Rival
Keyword(s):  
Line Segment ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Straight Line Segment ◽  
Line Segments ◽  
Small Circle ◽  
Partially Ordered ◽  
Straight Line ◽  
Finite Partially Ordered Set ◽  
Planar Lattices

A finite partially ordered set (poset) P is customarily represented by drawing a small circle for each point, with a lower than b whenever a < b in P, and drawing a straight line segment from a to b whenever a is covered by b in P (see, for example, G. Birkhoff [2, p. 4]). A poset P is planar if such a diagram can be drawn for P in which none of the straight line segments intersect.

ON INCIDENCE MODULO IDEAL RINGS

2007 ◽  
Vol 06 (04) ◽  
pp. 553-586 ◽  
Author(s):  
M. A. DOKUCHAEV ◽  
V. V. KIRICHENKO ◽  
B. V. NOVIKOV ◽  
A. P. PETRAVCHUK
Keyword(s):  
Linear Group ◽  
Partially Ordered Set ◽  
General Linear Group ◽  
Associative Ring ◽  
Ordered Set ◽  
General Linear ◽  
Local Rings ◽  
Partially Ordered ◽  
Finite Partially Ordered Set

For a given associative ring B, a two-sided ideal J ⊂ B and a finite partially ordered set P, we study the ring A = I(P, B, J) of incidence modulo J matrices determined by P. The properties of A involving its radical and quiver are investigated, and the interaction of A with serial rings is explored. The category of A-modules is studied if P is linearly ordered. Applications to the general linear group over some local rings are given.

scholarly journals Gelfand―Tsetlin Polytopes and Feigin―Fourier―Littelmann―Vinberg Polytopes as Marked Poset Polytopes

2011 ◽  
Vol DMTCS Proceedings vol. AO,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Thomas Bliem ◽  
Dido Salazar
Keyword(s):  
Representation Theory ◽  
Lie Algebra ◽  
Lie Algebras ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Ehrhart Polynomial ◽  
International Audience ◽  
Order Polytope ◽  
Finite Partially Ordered Set ◽  
The Relationship

International audience Stanley (1986) showed how a finite partially ordered set gives rise to two polytopes, called the order polytope and chain polytope, which have the same Ehrhart polynomial despite being quite different combinatorially. We generalize his result to a wider family of polytopes constructed from a poset P with integers assigned to some of its elements. Through this construction, we explain combinatorially the relationship between the Gelfand–Tsetlin polytopes (1950) and the Feigin–Fourier–Littelmann–Vinberg polytopes (2010, 2005), which arise in the representation theory of the special linear Lie algebra. We then use the generalized Gelfand–Tsetlin polytopes of Berenstein and Zelevinsky (1989) to propose conjectural analogues of the Feigin–Fourier–Littelmann–Vinberg polytopes corresponding to the symplectic and odd orthogonal Lie algebras. Stanley (1986) a montré que chaque ensemble fini partiellement ordonné permet de définir deux polyèdres, le polyèdre de l'ordre et le polyèdre des cha\^ınes. Ces polyèdres ont le même polynôme de Ehrhart, bien qu'ils soient tout à fait distincts du point de vue combinatoire. On généralise ce résultat à une famille plus générale de polyèdres, construits à partir d'un ensemble partiellement ordonné ayant des entiers attachés à certains de ses éléments. Par cette construction, on explique en termes combinatoires la relation entre les polyèdres de Gelfand-Tsetlin (1950) et ceux de Feigin-Fourier-Littelmann-Vinberg (2010, 2005), qui apparaissent dans la théorie des représentations des algèbres de Lie linéaires spéciales. On utilise les polyèdres de Gelfand-Tsetlin généralisés par Berenstein et Zelevinsky (1989) afin d'obtenir des analogues (conjecturés) des polytopes de Feigin-Fourier-Littelmann-Vinberg pour les algèbres de Lie symplectiques et orthogonales impaires.

The superintuitionistic predicate logic of finite Kripke frames is not recursively axiomatizable

2005 ◽  
Vol 70 (2) ◽  
pp. 451-459 ◽  
Author(s):  
Dmitrij Skvortsov
Keyword(s):  
Partially Ordered Set ◽  
Possible Worlds ◽  
Ordered Set ◽  
Predicate Logic ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Kripke Frames ◽  
Finite Partially Ordered Set

AbstractWe prove that an intermediate predicate logic characterized by a class of finite partially ordered sets is recursively axiomatizable iff it is “finite”, i.e., iff it is characterized by a single finite partially ordered set. Therefore, the predicate logic LFin of the class of all predicate Kripke frames with finitely many possible worlds is not recursively axiomatizable.

scholarly journals Essential ideals of incidence algebras

2000 ◽  
Vol 68 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Eugene Spiegel
Keyword(s):  
Commutative Ring ◽  
Left Ideal ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Locally Finite ◽  
Partially Ordered ◽  
Incidence Algebra ◽  
Incidence Algebras ◽  
Essential Ideal ◽  
Finite Partially Ordered Set

AbstractIt is determined when there exists a minimal essential ideal, or minimal essential left ideal, in the incidence algebra of a locally finite partially ordered set defined over a commutative ring. When such an ideal exists, it is described.

Decidability and ℵ0-categoricity of theories of partially ordered sets

1980 ◽  
Vol 45 (3) ◽  
pp. 585-611 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Linear Order ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Ordered Sets ◽  
Categorical Theory ◽  
Partially Ordered ◽  
Decidable Theory ◽  
Finite Partially Ordered Set

AbstractThis paper is primarily concerned with ℵ0-categoricity of theories of partially ordered sets. It contains some general conjectures, a collection of known results and some new theorems on ℵ0-categoricity. Among the latter are the following.Corollary 3.3. For every countable ℵ0-categoricalthere is a linear order of A such that (, <) is ℵ0-categorical.Corollary 6.7. Every ℵ0-categorical theory of a partially ordered set of finite width has a decidable theory.Theorem 7.7. Every ℵ0-categorical theory of reticles has a decidable theory.There is a section dealing just with decidability of partially ordered sets, the main result of this section beingTheorem 8.2. If (P, <) is a finite partially ordered set and KP is the class of partially ordered sets which do not embed (P, <), then Th(KP) is decidable iff KP contains only reticles.

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