Partially ordered sets and the independence property

1989 ◽  
Vol 54 (2) ◽  
pp. 396-401
Author(s):  
James H. Schmerl
Keyword(s):  
Distributive Lattice ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Finite Width ◽  
Order Property ◽  
Ordered Sets ◽  
Independence Property ◽  
Partially Ordered ◽  
Finite Dimensional

AbstractNo theory of a partially ordered set of finite width has the independence property, generalizing Poizat's corresponding result for linearly ordered sets. In fact, a question of Poizat concerning linearly ordered sets is answered by showing, moreover, that no theory of a partially ordered set of finite width has the multi-order property. It then follows that a distributive lattice is not finite-dimensional iff its theory has the independence property iff its theory has the multi-order property.

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.

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 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.

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).

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.

scholarly journals A remark on the extension of the concept of incidence algebras to nonlocally finite partially ordered sets

2004 ◽  
Vol 2004 (40) ◽  
pp. 2145-2147
Author(s):  
Boniface I. Eke
Keyword(s):  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Ordered Sets ◽  
Partially Ordered ◽  
Incidence Algebra ◽  
Incidence Algebras ◽  
Finite Partially Ordered Set

An incidence algebra of a nonlocally finite partially ordered setQis a very rare concept, perhaps nonexistent. In this note, we will attempt to construct such an algebra.

Free Lattices Generated by Partially Ordered Sets and Preserving Bounds

1964 ◽  
Vol 16 ◽  
pp. 136-148 ◽  
Author(s):  
R. A. Dean
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Partially Ordered Set ◽  
Ordered Set ◽  
Partially Ordered Sets ◽  
Free Lattice ◽  
Ordered Sets ◽  
Partially Ordered

A construction of the free lattice generated by a partially ordered set P and preserving every least upper bound (lub) and greatest lower bound (glb) of pairs of elements existing in P has been given by Dilworth (2, pp. 124-129) and, when P is finite, by Gluhov (5).The results presented here construct the free lattice FL generated by the partially ordered set P and preserving(1) the ordering of P(2) those lub's of a family of finite subsets of P which possess lub's in P

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