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 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.

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.

scholarly journals DECISION AND SEPARABILITY PROBLEMS FOR BAUMSLAG–SOLITAR SEMIGROUPS

2002 ◽  
Vol 12 (01n02) ◽  
pp. 33-49 ◽  
Author(s):  
DAVID A. JACKSON
Keyword(s):  
Finite Groups ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Residually Finite ◽  
Partially Ordered ◽  
Finite Partially Ordered Set

We show that the semigroups Sk,ℓ having semigroup presentations <a, b:abk = bℓ a> are residually finite and finitely separable. Generally, these semigroups have finite separating images which are finite groups and other finite separating images which are semigroups of order-increasing transformations on a finite partially ordered set. These semigroups thus have vastly different residual and separability properties than the Baumslag–Solitar groups which contain them.

scholarly journals TWO APPROACHES TO MÖBIUS INVERSION

2011 ◽  
Vol 85 (1) ◽  
pp. 68-78
Author(s):  
I-CHIAU HUANG
Keyword(s):  
Partially Ordered Set ◽  
Inversion Formula ◽  
Ordered Set ◽  
Lagrange Inversion ◽  
Locally Finite ◽  
Partially Ordered ◽  
Möbius Inversion ◽  
Möbius Inversion Formula ◽  
Schauder Bases ◽  
Finite Partially Ordered Set

AbstractThe Möbius inversion formula for a locally finite partially ordered set is realized as a Lagrange inversion formula. Schauder bases are introduced to interpret Möbius inversion.

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