Order Isomorphism Research Articles

Order isomophisms between Riesz spaces

The first aim of this paper is to give a description of the (not necessarily linear) order isomorphisms C(X)rightarrow C(Y) where X, Y are compact Hausdorff spaces. For a simple case, suppose X is metrizable and T is such an order isomorphism. By a theorem of Kaplansky, T induces a homeomorphism tau :Xrightarrow Y. We prove the existence of a homeomorphism Xtimes mathbb {R}rightarrow Ytimes mathbb {R} that maps the graph of any fin C(X) onto the graph of Tf. For nonmetrizable spaces the result is similar, although slightly more complicated. Secondly, we let X and Y be compact and extremally disconnected. The theory of the first part extends directly to order isomorphisms C^{infty }(X)rightarrow C^{infty }(Y). (Here C^{infty }(X) is the space of all continuous functions Xrightarrow [-infty ,infty ] that are finite on a dense set.) The third part of the paper considers order isomorphisms T between arbitrary Archimedean Riesz spaces E and F. We prove that such a T extends uniquely to an order isomorphism between their universal completions. (In the absence of linearity this is not obvious.) It follows, that there exist an extremally disconnected compact Hausdorff space X, Riesz isomorphisms hat{} of E and F onto order dense Riesz subspaces of C^{infty }(X) and an order isomorphism S:C^{infty }(X)rightarrow C^{infty }(X) such that hat{Tf}=Shat{f} (fin E).

Pure patterns of order 2

We provide mutual elementary recursive order isomorphisms between classical ordinal notations, based on Skolem hulling, and notations from pure elementary patterns of resemblance of order 2, showing that the latter characterize the proof-theoretic ordinal 1∞ of the fragment Π11–CA0 of second order number theory, or equivalently the set theory KPℓ0. As a corollary, we prove that Carlson's result on the well-quasi orderedness of respecting forests of order 2 implies transfinite induction up to the ordinal 1∞. We expect that our approach will facilitate analysis of more powerful systems of patterns.

Order Isomorphisms of Operator Intervals

We develop a general theory of order isomorphisms of operator intervals. In this way we unify and extend several known results, among others the famous Ludwig’s description of ortho-order automorphisms of effect algebras and Molnar’s characterization of bijective order preserving maps on bounded observables. Besides proving several new results, one of the main contributions of the paper is to provide self-contained proofs of several known theorems whose original proofs depend on various deep results from functional analysis, operator algebras, and geometry. At the end we will show the optimality of the obtained theorems using Lowner’s theory of operator monotone functions.

The parabolic algebra on Lp spaces

The parabolic algebra was introduced by Katavolos and Power, in 1997, as the SOT-closed operator algebra acting on L2(R) that is generated by the translation and multiplication semigroups. In particular, they proved that this algebra is reflexive and is equal to the Fourier binest algebra, that is, to the algebra of operators that leave invariant the subspaces in the Volterra nest and its analytic counterpart. We prove that a similar result holds for the corresponding algebras acting on Lp(R), where 1<p<∞. In the last section, it is also shown that the reflexive closures of the Fourier binests on Lp(R) are all order isomorphic for 1<p<∞.

Close hereditary C*-subalgebras and the structure of quasi-multipliers

We answer a question of Takesaki by showing that the following can be derived from the thesis of Shen: if A and B are σ-unital hereditary C*-subalgebras of C such that ‖p – q‖ &lt; 1, where p and q are the corresponding open projections, then A and B are isomorphic. We give some further elaborations and counterexamples with regard to the σ-unitality hypothesis. We produce a natural one-to-one correspondence between complete order isomorphisms of C*-algebras and invertible left multipliers of imprimitivity bimodules. A corollary of the above two results is that any complete order isomorphism between σ-unital C*-algebras is the composite of an isomorphism with an inner complete order isomorphism. We give a separable counterexample to a question of Akemann and Pedersen; namely, the space of quasi-multipliers is not linearly generated by left and right multipliers. But we show that the space of quasi-multipliers is multiplicatively generated by left and right multipliers in the σ-unital case. In particular, every positive quasi-multiplier is of the form T*T for T a left multiplier. We show that a Lie theory consequence of the negative result just stated is that the map sending T to T*T need not be open, even for very nice C*-algebras. We show that surjective maps between σ-unital C*-algebras induce surjective maps on left, right, and quasi-multipliers. (The more significant similar result for multipliers is Pedersen's non-commutative Tietze extension theorem.) We elaborate the relations of the above with continuous fields of Hilbert spaces and in so doing answer a question of Dixmier and Douady. We discuss the relationship of our results to the theory of perturbations of C*-algebras.

Diffusion determines the compact manifold

We provide a short proof for the theorem that two compact Riemannian manifolds are isomorphic if and only there exists an order isomorphism which intertwines between the heat semigroups on the manifolds.

Star order on operator and function algebras and its nonlinear preservers

The aim of this paper is to study the star order on operator and function algebras. It is shown that the infimum problem and the supremum problem on algebras of continuous functions have negative answer in general. Furthermore, we give a description of certain nonlinear star order isomorphisms between *-algebras. Finally, we describe general star order isomorphisms on normal parts of matrix algebras and atomic von Neumann algebras.

Orthogonal Measures on State Spaces and Context Structure of Quantum Theory

An interplay between recent topos theoretic approach and standard convex theoretic approach to quantum theory is discovered. Combining new results on isomorphisms of posets of all abelian subalgebras of von Neumann algebras with classical Tomita’s theorem from state space Choquet theory, we show that order isomorphisms between the sets of orthogonal measures (resp. finitely supported orthogonal measures) on state spaces endowed with the Choquet order are given by Jordan ∗-isomorphims between corresponding operator algebras. It provides new complete Jordan invariants for σ-finite von Neumann algebras in terms of decompositions of states and shows that one can recover physical system from associated structure of convex decompositions (discrete or continuous) of a fixed state.

Colour Visualisation of the Phase of Complex Signals

We propose a colour-based technique for the visualisation of the phase of complex signals, in particular of Fourier transforms and of analytic signals. Using the fact that both the hue and the angle are cyclic magnitudes, a one-to-one relationship can be established between angles and colour hues that respects the cyclic order: an order isomorphism. With the help of a Matlab GUI tool, called a chromatic profiler, the correspondence is made so that the cardinal points [2] of the cyclic variables are preserved, and the rest of the points are mapped so that the result is best readable. What the technique may lack in accuracy is gained in readability.

On derivations and their fixed point sets in residuated lattices

The main goal of this paper is to investigate derivations in residuated lattices and characterize some special types of residuated lattices in terms of derivations. In the paper, we discuss related properties of some particular derivations and give some characterizations of (good) ideal derivations. Then we obtain that the fixed point set of good ideal derivations is still a residuated lattice. Also, we prove that the set of all ideal derivation filters on residuated lattices with good ideal derivations can form a complete Heyting algebra, which is isomorphic to the lattice of all filters in the fixed point set for good ideal derivations. Moreover, we study principal ideal derivations and their adjoint derivations. Finally, we get that the fixed point set of principal ideal derivations and that of their adjoint derivations are order isomorphism. In particular, using the fixed point set of principal ideal derivations, we give characterizations of Heyting algebras and linearly ordered Heyting algebras.

Nonlinear order isomorphisms on function spaces

Let X be a topological space. A subset of C(X), the space of continuous real-valued functions on X, is a partially ordered set in the pointwise order. Suppose that X and Y are topological spaces, and A(X) and A(Y ) are subsets of C(X) and C(Y ) respectively. We consider the general problem of characterizing the order isomorphisms (order preserving bijections) between A(X) and A(Y ). Under some general assumptions on A(X) and A(Y ), and when X and Y are compact Hausdorff, it is shown that existence of an order isomorphism between A(X) and A(Y ) gives rise to an associated homeomorphism between X and Y . This generalizes a classical result of Kaplansky concerning linear order isomorphisms between C(X) and C(Y ) for compact Hausdorff X and Y . The class of near vector lattices is introduced in order to extend the result further to noncompact spaces X and Y . The main applications lie in the case when X and Y are metric spaces. Looking at spaces of uniformly continuous functions, Lipschitz functions, little Lipschitz functions, spaces of differentiable functions, and the bounded, “local” and “bounded local” versions of these spaces, characterizations of when spaces of one type can be order isomorphic to spaces of another type are obtained.

Open Set Lattices of Subspaces of Spectrum Spaces

AbstractWe take a unified approach to study the open set lattices of various subspaces of the spectrum of a multiplicative lattice L. The main aim is to establish the order isomorphism between the open set lattice of the respective subspace and a sub-poset of L. The motivating result is the well known fact that the topology of the spectrum of a commutative ring R with identity is isomorphic to the lattice of all radical ideals of R. The main results are as follows: (i) for a given nonempty set S of prime elements of a multiplicative lattice L, we define the S-semiprime elements and prove that the open set lattice of the subspace S of Spec(L) is isomorphic to the lattice of all S-semiprime elements of L; (ii) if L is a continuous lattice, then the open set lattice of the prime spectrum of L is isomorphic to the lattice of all m-semiprime elements of L; (iii) we define the pure elements, a generalization of the notion of pure ideals in a multiplicative lattice and prove that for certain types of multiplicative lattices, the sub-poset of pure elements of L is isomorphic to the open set lattice of the subspace Max(L) consisting of all maximal elements of L.

Persistence of Banach lattices under nonlinear order isomorphisms

Ordered vector spaces E and F are said to be order isomorphic if there is a (not necessarily linear) bijection $$T: E\rightarrow F$$ such that $$x\ge y$$ if and only if $$Tx \ge Ty$$ for all $$x,y \in E$$ . We investigate some situations under which an order isomorphism between two Banach lattices implies the persistence of some linear lattice structure. For instance, it is shown that if a Banach lattice E is order isomorphic to C(K) for some compact Hausdorff space K, then E is (linearly) isomorphic to C(K) as a Banach lattice. Similar results hold for Banach lattices order isomorphic to $$c_{0}$$ , and for Banach lattices that contain a closed sublattice order isomorphic to $$c_{0}$$ .

Classifying Finite-Dimensional C*-Algebras by Posets of Their Commutative C*-Subalgebras

We consider the functor C that to a unital C*-algebra A assigns the partial order set C(A) of its commutative C*-subalgebras ordered by inclusion. We investigate how some C*-algebraic properties translate under the action of C to order-theoretical properties. In particular, we show that A is finite dimensional if and only C(A) satisfies certain chain conditions. We eventually show that if A and B are C*-algebras such that A is finite dimensional and C(A) and C(B) are order isomorphic, then A and B must be *-isomorphic.

Intervals of permutations with a fixed number of descents are shellable

The set of all permutations, ordered by pattern containment, is a poset. We present an order isomorphism from the poset of permutations with a fixed number of descents to a certain poset of words with subword order. We use this bijection to show that intervals of permutations with a fixed number of descents are shellable, and we present a formula for the Möbius function of these intervals. We present an alternative proof for a result on the Möbius function of intervals [1,π] such that π has exactly one descent. We prove that if π has exactly one descent and avoids 456123 and 356124, then the intervals [1,π] have no nontrivial disconnected subintervals; we conjecture that these intervals are shellable.

Topologies Generated by Nested Collections

We study binary relations and topologies induced by means of nested collections of subsets of a given nonempty set. Dually, given a topological space, we characterize properties of the topology in terms of suitable nested families of open subsets. By means of this relationship between nested collections of sets and nested topologies, we study the representability of total preorders via (semi-)continuous real-valued order isomorphisms.

The “M” in digital elevation models

The “M” in digital elevation models (DEM) stands for model, which literally means “a schematic description of a system, theory, or phenomenon that accounts for its known or inferred properties and may be used for further study of its characteristics.” A DEM fulfills the requirement of “a schematic description” of terrain. However, how to make it account for the “known or inferred properties” warrants further scrutiny. This article outlines three properties of terrain and examines their four implications to DEM generation. The three properties are as follows: (1) each terrain point has a single, fixed elevation; (2) terrain points have an order and sequence that is determined by their elevations; and (3) terrain has skeletons. The four implications to DEM generation methods are as follows: (1) a method must be a bijection; (2) a method must be an isomorphism in order to preserve elevation sequence; (3) a method must guarantee that the vertical error at any point, not just checkpoints, is acceptable in order to assure the vertical accuracy of a DEM; and (4) a method must involve generalization if terrain skeletons are to be preserved. These implications are discussed in the context of light detection and ranging-derived DEMs. Generalization is highlighted as the top priority for future research.

On the Nonexistence of Order Isomorphisms between the Sets of All Self-Adjoint and All Positive Definite Operators

We prove that there is no bijective map between the set of all positive definite operators and the set of all self-adjoint operators on a Hilbert space with dimension greater than 1 which preserves the usual order (the one coming from the concept of positive semidefiniteness) in both directions. We conjecture that a similar assertion is true for general noncommutativeC*-algebras and present a proof in the finite dimensional case.

Diffusion determines the recurrent graph

We consider diffusion on discrete measure spaces as encoded by Markovian semigroups arising from weighted graphs. We study whether the graph is uniquely determined if the diffusion is given up to order isomorphism. If the graph is recurrent then the complete graph structure and the measure space are determined (up to an overall scaling). As shown by counterexamples this result is optimal. Without the recurrence assumption, the graph still turns out to be determined in the case of normalized diffusion on graphs with standard weights and in the case of arbitrary graphs over spaces in which each point has the same mass.These investigations provide discrete counterparts to studies of diffusion on Euclidean domains and manifolds initiated by Arendt and continued by Arendt/Biegert/ter Elst and Arendt/ter Elst.A crucial step in our considerations shows that order isomorphisms are actually unitary maps (up to a scaling) in our context.

Fuzzy Acts over Fuzzy Semigroups and Sheaves

Although fuzzy set theory and sheaf theory have been developed and studied independently, Ulrich Hohle shows that a large part of fuzzy set theory is in fact a subeld of sheaf theory. Many authors have studied math- ematical structures, in particular, algebraic structures, in both categories of these generalized (multi)sets. Using Hohle's idea, we show that for a (universal) algebra A, the set of fuzzy algebras over A and the set of subalgebras of the constant sheaf of algebras over A are order isomorphic. Then, among other things, we study the category of fuzzy acts over a fuzzy semigroup, so to say, with its universal algebraic as well as classic algebraic denitions.