scholarly journals Quadratic Gröbner bases arising from partially ordered sets

2017 ◽  
Vol 121 (1) ◽  
pp. 19 ◽  
Author(s):  
Takayuki Hibi ◽  
Kazunori Matsuda ◽  
Akiyoshi Tsuchiya
Keyword(s):  
Convex Hull ◽  
Ordered Set ◽  
Convex Polytope ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Toric Ideals ◽  
Partially Ordered ◽  
Initial Ideals ◽  
Order Polytope ◽  
Fano Polytope

The order polytope $\mathcal {O}(P)$ and the chain polytope $\mathcal {C}(P)$ associated to a partially ordered set $P$ are studied. In this paper, we introduce the convex polytope $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ which is the convex hull of $\mathcal {O}(P) \cup (-\mathcal {C}(Q))$, where both $P$ and $Q$ are partially ordered sets with $|P|=|Q|=d$. It will be shown that $\Gamma (\mathcal {O}(P), -\mathcal {C}(Q))$ is a normal and Gorenstein Fano polytope by using the theory of reverse lexicographic squarefree initial ideals of toric ideals.

scholarly journals Retracts of partially ordered sets

1979 ◽  
Vol 27 (4) ◽  
pp. 495-506 ◽  
Author(s):  
Dwight Duffus ◽  
Ivan Rival
Keyword(s):  
Fixed Points ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

AbstractLet P be a finite, connected partially ordered set containing no crowns and let Q be a subset of P. Then the following conditions are equivalent: (1) Q is a retract of P; (2) Q is the set of fixed points of an order-preserving mapping of P to P; (3) Q is obtained from P by dismantling by irreducibles.

Convex structures in a new kind of ordered fuzzy group1

2020 ◽  
Vol 39 (3) ◽  
pp. 4245-4257
Author(s):  
Hongping Liu ◽  
Ruiju Wei ◽  
Qian Ge
Keyword(s):  
Convex Hull ◽  
Binary Operation ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Convex Subgroup ◽  
Partially Ordered ◽  
Convex Structure ◽  
Fuzzy Group ◽  
Fuzzy Setting ◽  
Convex Spaces

By means of a fuzzy binary operation defined on partially ordered sets, a new kind of ordered fuzzy group is proposed in this paper. Some properties of this ordered fuzzy group are studied. Following that, its substructures, such as subgroup and convex subgroup, as well as its homomorphisms, along with their properties are explored. It is shown that each family of these substructures forms a convex structure, where the convex hull of a subset is exactly the (convex) subgroup generated by itself, and the homomorphisms between two ordered fuzzy groups are convexity-preserving mappings between the corresponding convex spaces. In addition, when these substructures are extended to fuzzy setting, several L-convex structures are constructed and investigated.

scholarly journals Some Common Fixed Point Theorems in Partially Ordered Sets

2020 ◽  
Vol 2020 ◽  
pp. 1-5
Author(s):  
Khadija Bouzkoura ◽  
Said Benkaddour
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Common Fixed Point ◽  
Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Common Fixed Point Theorems

The purpose of this paper is to prove some new fixed point theorem and common fixed point theorems of a commuting family of order-preserving mappings defined on an ordered set, which unify and generalize some relevant fixed point theorems.

CAYLEY GRAPHS OF PARTIALLY ORDERED SETS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250184 ◽  
Author(s):  
MOJGAN AFKHAMI ◽  
ZAHRA BARATI ◽  
KAZEM KHASHYARMANESH
Keyword(s):  
Cayley Graph ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Cayley Graphs ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Basic Properties ◽  
Vertex Set

In this paper, we introduce the Cayley graph of a partially ordered set (poset). Let (P, ≤) be a poset, and let S be a subset of P. We define the undirected Cayley graph of P, denoted by Cay (P, S), as a graph with vertex-set P and edge-set E consisting of those sets {x, y} such that y ∈ {x, s}ℓ or x ∈ {y, s}ℓ for some s ∈ S, where for a subset T of P, Tℓ is the set of all x ∈ P such that x ≤ t, for all t ∈ T. We study some basic properties of Cay (P, S) such as connectivity, diameter and girth.

The Ordering of Spec R

1976 ◽  
Vol 28 (4) ◽  
pp. 820-835 ◽  
Author(s):  
William J. Lewis ◽  
Jack Ohm
Keyword(s):  
Commutative Ring ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Prime Ideals ◽  
Ordered Sets ◽  
Partially Ordered

Let Specie denote the set of prime ideals of a commutative ring with identity R, ordered by inclusion; and call a partially ordered set spectral if it is order isomorphic to Spec R for some R. What are some conditions, necessary or sufficient, for a partially ordered set X to be spectral? The most desirable answer would be the type of result that would allow one to stare at the diagram of a given X and then be able to say whether or not X is spectral. For example, it is known that finite partially ordered sets are spectral (see [2] or [5]).

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.

Endomorphisms of Partially Ordered Sets

1998 ◽  
Vol 7 (1) ◽  
pp. 33-46
Author(s):  
DWIGHT DUFFUS ◽  
TOMASZ ŁUCZAK ◽  
VOJTĚCH RÖDL ◽  
ANDRZEJ RUCIŃSKI
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered

It is shown that every partially ordered set with n elements admits an endomorphism with an image of a size at least n1/7 but smaller than n. We also prove that there exists a partially ordered set with n elements such that each of its non-trivial endomorphisms has an image of size O((n log n)1/3).

A Theorem on Partially Ordered Sets, With Applications to Fixed Point Theorems

1961 ◽  
Vol 13 ◽  
pp. 78-82 ◽  
Author(s):  
Smbat Abian ◽  
Arthur B. Brown
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Fixed Point Theorems ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Prove Theorem ◽  
The Fixed Point Theorem ◽  
The Axiom Of Choice

In this paper the authors prove Theorem 1 on maps of partially ordered sets into themselves, and derive some fixed point theorems as corollaries.Here, for any partially ordered set P, and any mapping f : P → P and any point a ∈ P, a well ordered subset W(a) ⊂ P is constructed. Except when W(a) has a last element ε greater than or not comparable to f(ε), W(a), although constructed differently, is identical with the set A of Bourbaki (3) determined by a, f , and P1: {x|x ∈ P, x ≤ f(x)}.Theorem 1 and the fixed point Theorems 2 and 4, as well as Corollaries 2 and 4, are believed to be new.Corollaries 1 and 3 are respectively the well-known theorems given in (1, p. 54, Theorem 8, and Example 4).The fixed point Theorem 3 is that of (1, p. 44, Example 4); and has as a corollary the theorem given in (2) and (3).The proofs are based entirely on the definitions of partially and well ordered sets and, except in the cases of Theorem 4 and Corollary 4, make no use of any form of the axiom of choice.

Decidability and finite axiomatizability of theories of ℵ0-categorical partially ordered sets

1981 ◽  
Vol 46 (1) ◽  
pp. 101-120 ◽  
Author(s):  
James H. Schmerl
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Decidable Theory ◽  
Axiomatizable Theory ◽  
Finitely Axiomatizable Theory

AbstractEvery ℵ0-categorical partially ordered set of finite width has a finitely axiomatizable theory. Every ℵ0-categorical partially ordered set of finite weak width has a decidable theory. This last statement constitutes a major portion of the complete (with three exceptions) characterization of those finite partially ordered sets for which any ℵ0-categorical partially ordered set not embedding one of them has a decidable theory.

D-Continuity and Petri’s Axioms of Concurrency for Nonsequential Process Models

1987 ◽  
Vol 10 (2) ◽  
pp. 161-211
Author(s):  
Eike Best ◽  
Agathe Merceron
Keyword(s):  
Petri Nets ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Process Models ◽  
Sequential Process ◽  
Ordered Sets ◽  
Related Property ◽  
Real Numbers ◽  
Partially Ordered

A non-sequential process can be modelled by a partially ordered set. Conversely, one is led to study the properties to be fulfilled by a poset so that it can reasonably be viewed as the model of a non-sequential process. To this end, C.A. Petri has proposed a set of con currency axioms and a related property called D-continuity, a generalised version for partially ordered sets of Dedekind’s completeness property of the real numbers. In this paper we study Petri’s axioms of concurrency and some of their interdependencies. We also derive several characterisations of D-continuity and exhibit its relation with the axioms of concurrency. Furthermore we apply our work to Petri nets: we introduce occurrence nets, some special posets which model the processes of a system net and we present their relations to D-continuity and the axioms of con currency. Finally we identify the class of the system nets whose processes are D-continuous and satisfy the axioms of concurrency.

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