Partial orders on the types in $\beta N$

CHAPTER 19 - Versions of Martin's Axiom

We define a countable antichain condition (ccc) property for partial orderings, weaker than precalibre-@1, and show that Martin's axiom restricted to the class of partial orderings that have the prop- erty does not imply Martin's axiom for �-linked partial orderings. This yields a new solution to an old question of the first author about the relative strength of Martin's axiom for �-centered partial orderings to- gether with the assertion that every Aronszajn tree is special. We also answer a question of J. Steprans and S. Watson (1988) by showing that, by a forcing that preserves cardinals, one can destroy the precalibre-@1 property of a partial ordering while preserving its ccc-ness.

When newer compilations of decision support methods are examined, partial order methods as a central methodological aspect are rarely found. This is strange, since the role of multi-indicator systems is worldwide increasing. Multi-indicator systems induce in a natural manner partial orders. Why is partial order not seen as that important tool? The main reason appears to be that partially ordered sets are in general not completely ordered. There are incomparabilities, and incomparabilities do not lead to rankings. Thus, incomparabilities a priori hamper any decision. Approaching the decision from the need to get a unique ranking, it is clear that decision support systems aim toward deriving a one-dimensional scalar, by which a linear, i.e. a unique order can be derived. In the present paper an approach is selected to reduce the number of incomparabilities without losing the quality of insights, partially ordered sets allow. As an example economical relevant chemicals are studied.

It is well-known that on an inverse semigroup S the relation ≦ defined by a ≦ b if and only if aa−1 = ab−1 is a partial order (called the natural partial order) on S and that this relation is closely related to the global structure of S (cf. (1, §7.1), (10)). Our purpose here is to study a partial order on regular semigroups that coincides with the relation defined above on inverse semigroups. It is found that this relation has properties very similar to the properties of the natural partial order on inverse semigroups. However, this relation is not, in general, compatible with the multiplication in the semigroup. We show that this is true if and only if the semigroup is pseudo-inverse (cf. (8)). We also show how this relation may be used to obtain a simple description of the finest primitive congruence and the finest completely simple congruence on a regular semigroup.

Continuing the development [4] of an aspect of the approach to the axiomatization of quantum mechanics of G. W. Mackey [3], we consider here the real linear space X of signed measures on the set P of events generated by the states, and the set F0 of linear functionals on X which are induced in a natural way by the bounded observables. A necessary and sufficient condition for two events to be simultaneously measurable is found in terms of the order structure of F0, with the following consequence: if F0 is a lattice, P is deterministic. At the opposite extreme, F0 is said to be an anti-lattice1 if the greatest lower bound exists only for comparable pairs of its elements and we show in this case that the center of P is trivial. Our results extend those of R. V. Kadison [l], in which F0 and P are the self-adjoint operators and projections respectively in a uniformly closed self-adjoint operator algebra. While the framework and plan of the proofs were inspired by Kadison's work, almost none of the apparatus used by him is available here with the result that, in detail, our techniques are quite different from his. Let P be a weakly modular partially ordered set (see [4]). A function x from P to the non-negative real numbers and + oo is said to be a measure if x(0) =0 and x is countably additive in the sense that whenever {a,} is a pairwise orthogonal sequence of elements of P, then x(Ua?) = 2Zxiad- If x is a measure and {bt} EP is an increasing (decreasing) sequence with supremum (infimum) b, then x(&?) —*x(6). A countably additive function x from P to the extended real numbers is a signed measure if x(0) =0 and x takes on at most one of the values + =o and — oo ; x is finite if x(l) is finite. Define the functions 5 and i on the signed measures on P by six) = sup(x(a) : aEP\, fx) =inf {x(a): aEP} and set ||x|| =s(x)—?(x). Clearly ||x|| < oo if and only if x is finite. It iseasy toseethat||x|| = sup {x(a)— xia'):aE P} ■ Lemma 1. Let X be a real linear space of finite signed measures on P. Then the function || || defined above is a norm for X and, under this norm and its natural partial ordering, X is a partially ordered normed linear space. That is,2

j-2 - Consistency Results in Topology, I: Quotable Principles

Let V be any vector space and P(V) the set of all partial linear transformations defined on V, that is, all linear α: A → B, where A, B are subspaces of V. Then P(V) is a semigroup under composition, which is partially ordered by ⊆ (that is, α ⊆ β if and only if dom α ⊆ dom β and α = β | dom α). We compare this order with the so-called 'natural partial order' ≤ on P(V) and we determine their meet and join. We also describe all elements of P(V) that are minimal (or maximal) with respect to each of these four orders, and we characterize all elements that are 'compatible' with them. In addition, we answer similar questions for the semigroup T(V) consisting of all α ∈ P(V) whose domain equals V. Other orders have been defined by Petrich on any regular semigroup: three of them form a chain below ≤, and we show that two of these are equal on the semigroup P(V) and on the ring T(V). We also consider questions for these orders that are similar to those already mentioned

It is hard to find nontrivial positive partition relations which hold for many ordinals in ordinary set theory, or even ordinary set theory with the additional assumption of the Generalized Continuum Hypothesis.Erds, Hajnal and Milner have proved that limit ordinals a < u;"+ satisfy a positive partition relation that can be expressed in graph theoretic terms.In symbols one writes a - (a, infinite path)2 to mean that every graph on an ordinal a either has a subset order isomorphic to a in which no two points are joined by an edge or has an infinite path.This positive result generalizes to ordinals of cardinality Nm for m a natural number.However, the argument, based on a set mapping theorem, works only on the initial segment of the limit ordinals of cardinality Nm for which the set mapping theorem is true.In this paper, the Generalized Continuum Hypothesis is used to construct counterexamples for a cofinal set of ordinals of cardinality Nm, where m is a natural number at least two. Introduction.Erds, Hajnal and Milner [3] were the first to look at the partition relation a -> (a, infinite path)2 in their 1969 paper on set mappings.They were able to prove positive results for many ordinals, but their method using set mappings only worked for ordinals of relatively simple order structure.However, this particular relation looked most promising as a nontrivial one that many ordinals might satisfy.J. Larson [6] used Martin's Axiom to extend the positive result to limit ordinals less than the continuum.(It is not hard to construct counterexamples for sucessor ordinals.)However, under the assumption of the Generalized Continuum Hypothesis, above the continuum, counterexamples start to appear, as indicated in the statement of the main theorem below.THEOREM.Assume the Generalized Continuum Hypothesis.For every positive integer n > 2, there is a cofinal set of ordinals a < ojn so that a -*+ (a, infinite path)2.For ordinals of power continuum, the situation appears to be more delicate.In A diamond example of an ordinal graph with no infinite paths, J. Baumgartner and J. Larson [1] used Jensen's Diamond Principle to show that for all ordinals a with w"+2 < a < w2, the negative partition relation a -** (a, infinite path)2 holds.

We introduce a preorder on an inverse semigroup S associated to any normal inverse subsemigroup N, that lies between the natural partial order and Green’s $${\mathcal {J}}$$ –relation. The corresponding equivalence relation $$\simeq _N$$ is not necessarily a congruence on S, but the quotient set does inherit a natural ordered groupoid structure. We show that this construction permits the factorisation of any inverse semigroup homomorphism into a composition of a quotient map and a star-injective functor, and that this decomposition implies a classification of congruences on S. We give an application to the congruence and certain normal inverse subsemigroups associate to an inverse monoid presentation.

The absolute order on the symmetric group, constructible partially ordered sets and Cohen–Macaulay complexes

On a Class of Lattice Ordered Inverse Semigroups

Assuming the Axiom of Constructibility, necessary and sufficient conditions are given for the vanishing of Ext\ for rings A of global dimension 1.Using Martin's Axiom, the necessity of these conditions is shown not to be a theorem of ZFC.Applications are given to abelian group theory, including a partial solution (assuming V = L) to a problem of Baer on the splitting of abelian groups.Some independence results in abelian group theory are also proved.Introduction.In a remarkable recent paper [22], S. Shelah proved that Whitehead's problem is unsolvable in ordinary set theory (Zermelo-Frankel set theory with the Axiom of Choice).More precisely, he proved that two different answers to the problem are obtained when set theory is extended by adding two different axioms: the Axiom of Constructibility and Martin's Axiom.(See Theorem 4.1.)Shelah's arguments were specific to Whitehead's problem and nonhomological in character.In this paper we extend Shelah's techniques in order to obtain some general results about the vanishing of ExtA (for modules over a ring A of global dimension 1) in different models of set theory, and we apply them to some specific problems in abelian group theory.Assuming the Axiom of Constructibility we obtain necessary and sufficient conditions for the vanishing of ExtA (Theorems 1.2 and 1.5).To state the result in.a special case: if C and A are abelian groups of cardinality N,, Ext(C, A) -0 if and only if C is the union of an increasing chain of countable subgroups [Cv\v < «,} such that the chain is smooth (i.e., C0 -U"<oC, for aii iimit ordinals a) and satisfies Ext(C0, A) = 0 and E\i(Cv+x/Cv,A) = 0 for all v < K,.Although the sufficiency of this condition is an easy consequence of classical homological algebra (see Theorem 1.2), the proof of necessity makes use of the Axiom of Constructibility (Theorem 1.5).The necessity of this condition is not provable in ordinary set theory; in particular the condition is not necessary when Martin's Axiom is assumed ( §3).

Special types of coverings and axiomatization of rough sets based on partial orders

Mazurkiewicz trace theory is not powerful enough to describe concurrency paradigms as, for instance, the “Producer / Consumer”. We propose in this paper a generalization of Mazurkiewicz trace monoids which allows to model such problems. We consider quotients of the free monoids by congruences which preserve the commutative images of words. An equivalence class in the quotient monoid consists of all the sequential observations of a distributed computation. In order to characterize congruences which do model concurrency, we study the relationship of this approach and the classical representation of distributed computations with partial orders. We show that the only congruences for which the classes can be represented by partial orders and the concatenation transfers modularly to partial orders are congruences generated by commutations, that is trace congruences. We prove necessary conditions and sufficient conditions on congruences so that their classes can be represented by partial orders. In particular, an important sufficient condition covers both trace congruences and the “Producer / Consumer” congruence.

k-10 - Manifold