An Order Model for Infinite Classical States

In 2002 Coecke and Martin (Research Report PRG-RR-02-07, Oxford University Computing Laboratory,2002) created a model for the finite classical and quantum states in physics. This model is based on a type of ordered set which is standard in the study of information systems. It allows the information content of its elements to be compared and measured. Their work is extended to a model for the infinite classical states. These are the states which result when an observable is applied to a quantum system. When this extended order is restricted to a finite number of coordinates, the model of Coecke and Martin is obtained. The infinite model retains many desirable aspects of the finite model, such as pure states as maximal elements and expected behavior of thermodynamic entropy. But it looses some of the important domain theoretic aspects, such as having a least element and exactness. Shannon entropy is no longer defined over the entire model and both it and thermodynamic entropy cease to be a measurements in the sense of Martin

A Note on Reordering Ordered Topological Spaces and the Existence of Continuous, Strictly Increasing Functions

The origin of this paper is in a question that was asked of the author by Michael Wellman, a computer scientist who works in artificial intelligence at Wright Patterson Air Force Base in Dayton, Ohio. He wanted to know if, starting with Rn and its usual topology and product partial order, he could linearly reorder every finite subset and still obtain a continuous function from Rn into R that was strictly increasing with respect to the new order imposed on Rn. It is the purpose of this paper to explore the structural characteristics of ordered topological spaces which have this kind of behavior

Properties of Weak Domain Representable Spaces

We will explore some of the basic properties of weak domain representable (wdr) spaces, including hereditary properties and properties of products. In particular, we will construct a Baire space that is not wdr, show that products of wdr spaces are wdr, and demonstrate that the factors of a product that is wdr need not themselves be wdr. We will also show that if X is a wdr space and Y ⊆ X such that |Y|=|X| then Y is wdr. We can declare a subset of a wdr space X to be open or to consist of isolated points without losing the property of being wdr

Pixley-Roy Hyperspaces of ω-graphs

The techniques developed by Wage and Norden are used to show that the Pixley-Roy hyperspaces of any two ω-graphs are homeomorphic. The Pixley-Roy hyperspaces of several subsets of Rn are also shown to be homeomorphic

Dissertation: The Least Fixed Point Property for ω-chain Continuous Functions

The basic definitions are given in the first section, including those for ω-chain continuity, ω-chain completeness, and the least fixed point property for ω-chain continuous functions. Some of the relations between completeness and fixed point properties in partially ordered sets are stated and it is briefly shown how the question basic to the dissertation arises. In the second section, two examples are given showing that a partially ordered set need not be ω-chain complete to have the least fixed point property for ω-chain continuous functions. Retracts are discussed in section 3, where it is seen that they are not sufficient to characterize those partially ordered sets having the least fixed point property for ω-chain continuous functions. In section 4, the relation between finite width and the least fixed point property for ω-chain continuous functions is explored. Section 5 introduces the notion of a layered partially ordered set and discusses some of its problems

Three Counterexamples Concerning ω-Chain Continuous Functions and Fixed-point Properties

A partially ordered set is ω-chain complete if, for every countable chain, or ω-chain, in P, the least upper bound of C, denoted by sup C, exists. Notice that C could be empty, so an ω-chain complete partially ordered set has a least element, denoted by 0

Sobriety in Delta not Sober

We will show that the space delta not sober defined by Coecke and Martin is sober in the Scott topology, but not in the weakly way below topology

Linearly Ordered Topological Spaces and Weak Domain Representability

It is well known that domain representable spaces, that is topological spaces that are homeomorphic to the space of maximal elements of some domain, must be Baire. In this paper it is shown that every linearly ordered topological space (LOTS) is homeomorphic to an open dense subset of a weak domain representable space. This means that weak domain representable spaces need not be Baire

A Note on Irreducibility and Weak Covering Properties

A space X is irreducible if every open cover of X has a minimal open refinement. Interest in irreducibility began when Arens and Dugendji used this property to show that metacompact countably compact spaces are compact. It was natural, then, to find out what other types of spaces would be irreducible and therefore compact in the presence of countable compactness or Lindelof in the presence of N1-compactness. … It is shown in this paper that T1 δθ -refinable spaces and T1 weakly δθ-refinable spaces are irreducible. Since examples of Lindelof spaces that are neither T1 nor irreducible can be easily constructed, it is clear that the spaces must be T1

The Least Fixed Point Property for ω-chain Continuous Functions

A partially ordered set P is ω-chain complete if every countable chain (including the empty set) in P has a supremum. … Notice that an ω-chain continuous function must preserve order. P has the (least) fixed point property for ω-chain continuous functions if every ω-chain continuous function from P to itself has (least) fixed point. It has been shown that a partially ordered set does not have to be ω-chain complete to have the least fixed point property for ω-chain continuous functions. This answers a question posed by G. Plotkin in 1978. I.I. Kolodner has shown that an ω-chain complete partially ordered set has the least fixed point property for ω-chain continuous functions. Plotkin and Smythe and others have used ω-chain complete partially ordered sets in their study of models for theoretical computer science in order to have fixed or least fixed point properties. The result should also be compared with G. Markowsky’s result that to have the least fixed point property (every order preserving function has a least fixed point), a partially ordered set must be chain complete. It is the purpose of this paper to look at some cases in which ω-chain completeness and the least fixed point property for ω-chain continuous functions are equivalent