Order Isomorphism Research Articles

Order isomorphisms on unbounded self-adjoint operators

We provide a comprehensive description of all order isomorphisms between several types of unbounded self-adjoint operator sets. Namely, sets of all positive operators, sets of all positive boundedly invertible operators, and those of all self-adjoint operators.

Order isomorphisms between real little Lipschitz algebras with Lipschitz involution

Order isomorphisms between real little Lipschitz algebras with Lipschitz involution

On the order types of hammocks for domestic string algebras

In the representation-theoretic study of finite dimensional associative algebras over an algebraically closed field, Brenner introduced certain partially ordered sets known as hammocks to encode factorizations of maps between indecomposable finitely generated modules. In the context of domestic string algebras, Schröer introduced a simpler version of hammocks in his doctoral thesis that are bounded discrete linear orders. In this paper, we characterize the class of order types(=order isomorphism classes) of hammock linear orders for domestic string algebras as the bounded discrete ones amongst the class LOfp of finitely presented linear orders–the smallest class of linear orders containing finite linear orders as well as ω, and that is closed under isomorphisms, order reversal, finite order sums and antilexicographic products.In fact, we provide a multi-step algorithm to compute the order type of any closed interval in the hammock, and prove the correctness of this algorithm. A major step of this algorithm is the construction of a variation, which we call the arch bridge quiver, of a finite combinatorial gadget called the bridge quiver introduced by Schröer. He utilised the graph-theoretic properties of the bridge quiver for the computation of some representation-theoretic numerical invariants of domestic string algebras. The vertices of the bridge quiver are (representatives of cyclic permutations of) bands and its arrows are certain band-free strings. There is a natural but ill-behaved partial binary operation, ∘, on a superset of the set of bridges consisting of weak bridges such that bridges are precisely the ∘-irreducibles. We equip an even larger yet finite set of weak arch bridges with another partial binary operation, ∘H, to obtain a finite category. The binary operation ∘H uses isomorphisms between hammocks and explicitly relies on the description of the domestic string algebra as a bound quiver algebra. Each weak arch bridge admits a unique ∘H-factorization into arch bridges, i.e., the ∘H-irreducibles.

An Alternative Proof for the Expected Number of Distinct Consecutive Patterns in a Random Permutation

Let $\pi_n$ be a uniformly chosen random permutation on $[n]$. Using an analysis of the probability that two overlapping consecutive $k$-permutations are order isomorphic, the authors of a recent paper showed that the expected number of distinct consecutive patterns of all lengths $k\in\{1,2,\ldots,n\}$ in $\pi_n$ is $\frac{n^2}{2}(1-o(1))$ as $n\to\infty$. This exhibited the fact that random permutations pack consecutive patterns near-perfectly. We use entirely different methods, namely the Stein-Chen method of Poisson approximation, to reprove and slightly improve their result.

Order type Tingley's problem for type I finite von Neumann algebras

Let M and N be von Neumann algebras, S(M)+ be the collection of all positive norm one elements in M, and PM be the projection lattice of M. Let Φ:PM→PN be a metric preserving order isomorphism and Λ:S(M)+→S(N)+ be a bijective isometry. When both M and N are type I finite, we establish that the map Φ extends to a Jordan ⁎-isomorphism from M onto N. On the other hand, if M and N are of the form ⨁n=1k0Mn, where Mn is either zero or a von Neumann algebra of type In, then the map Λ extends to a Jordan ⁎-isomorphism from M onto N. On our way, we also verify that when M and N are general von Neumann algebras, the map Λ extends to a Jordan ⁎-isomorphism if and only if Λ|PM∖{0} is a bi-orthogonality preserving bijection from PM∖{0} onto PN∖{0}.

Galois connections and isomorphism of simultaneous ordered relations

In order theory, partially ordered sets are only equipped with one relation which decides the entire structure/Hasse diagram of the set. In this paper we have presented how partially ordered sets can be studied under simultaneous partially ordered relations which we have called binary posets. The paper is motivated by the problem of operating a set simultaneously under two distinct partially ordered relations. It has been shown that binary posets follow duality principle just like posets do. Within this framework, some new definitions concerning maximal and minimal elements are also presented. Furthermore, some theorems on order isomorphism and Galois connections are derived.

Order isomorphisms on effect algebras of the C⁎-algebras of type [formula omitted

Let A be a unital C⁎-algebra equipped with its natural order. As usual the effect algebra of A is the interval {a∈A:0≤a≤IA}, where IA denotes the unit of A. In this paper, we give a complete description of order isomorphisms between effect algebras of C⁎-algebras of type C(X)⊗B(H), where C(X) stands for the algebra of all continuous complex valued functions on a (non pathological) Hausdorff compact space X and B(H) denotes the algebra of all bounded linear operators on a complex Hilbert space H. Our results generalize some works by L. Molnár and P. Šemrl.

Dynamic Student Data Management Using Resource Optimization Technology in Higher Education Platforms

This study combines intelligent resource optimization technology to build a dynamic student data management model and suggests a fuzzy hierarchical network representation model based on isomorphism and homogeneity in order to increase the effectiveness of dynamic student data management in colleges and universities. The important semantics of nodes in the network are also captured in this study using fuzzy k-kernel decomposition as a technique for multigranularity partitioning. Based on the SIR model, FHNE compares the production of the sequence to the process of information transmission in the random walk stage, which increases the node sequence’s accuracy. According to the research, the dynamic student data management system that is used in the higher education platform that is suggested in this study can significantly increase the effectiveness of managing student data.

Interpolation Pseudo-Ordered Rings

Characteristics of partially pseudo-ordered (K-ordered) rings are considered. Properties of the set L(R) of all convex directed ideals in pseudo-ordered rings are described. The convexity of ideals has the meaning of the Abelian convexity, which is based on the definition of a convex subgroup for a partially ordered group. It is proved that if R is an interpolation pseudo-ordered ring, then, in the lattice L(R), the union operation is completely distributive with respect to the intersection. Properties of the lattice L(R) for pseudo-lattice pseudo-ordered rings are investigated. The second and third theorems of ring order isomorphisms for interpolation pseudo-ordered rings are proved. Some theorems are proved for principal convex directed ideals of interpolation pseudo-ordered rings. The principal convex directed ideal Ia of a partially pseudo-ordered ring R is the smallest convex directed ideal of the ring R that contains the element a ∈ R. The analog for the third theorem of ring order isomorphisms for principal convex directed ideals is demonstrated for interpolation pseudo-ordered rings.

Ergodic decompositions of Dirichlet forms under order isomorphisms

We study ergodic decompositions of Dirichlet spaces under intertwining via unitary order isomorphisms. We show that the ergodic decomposition of a quasi-regular Dirichlet space is unique up to a unique isomorphism of the indexing space. Furthermore, every unitary order isomorphism intertwining two quasi-regular Dirichlet spaces is decomposable over their ergodic decompositions up to conjugation via an isomorphism of the corresponding indexing spaces.

The expected number of distinct consecutive patterns in a random permutation

Abstract Let π n be a uniformly chosen random permutation on [n]. Using an analysis of the probability that two overlapping consecutive k-permutations are order isomorphic, we show that the expected number of distinct consecutive patterns of all lengths k ∈ {1, 2,…, n} in π n is n 2 2 ( 1 - o ( 1 ) ) {{{n^2}} \over 2}\left( {1 - o\left( 1 \right)} \right) as n → ∞. This exhibits the fact that random permutations pack consecutive patterns near-perfectly.

On Order Isomorphisms Intertwining Semigroups for Dirichlet Forms

This paper is devoted to characterizing so-called order isomorphisms intertwining the \(L^2\)-semigroups of two quasi-regular Dirichlet forms. We first show that every unitary order isomorphism intertwining semigroups is the composition of h-transformation and quasi-homeomorphism. In addition, under the assumption that the underlying spaces admit so-called irreducible decompositions for Dirichlet forms, every (not necessarily unitary) order isomorphism intertwining semigroups can be expressed as the composition of h-transformation, quasi-homeomorphism and multiplication by a certain step function.

Isomorphisms of spectral lattices

The paper deals with spectral order isomorphisms between certain spectral sublattices of direct sums of $$AW$$ *-factors. We prove that these maps consist of spectral order isomorphisms between spectral sublattices of individual direct summands. Consequently, we obtain a complete description of spectral order isomorphisms in the case of atomic $$AW$$ *-algebras. This includes the setting of matrix algebras. Moreover, we also exhibit the general form of spectral order orthoisomorphisms between various spectral sublattices of direct sums of $$AW$$ *-factors.

An Improved Order-Preserving Pattern Matching Algorithm Using Fingerprints

Two strings of the same length are order isomorphic if their relative orders are the same. The order-preserving pattern matching problem is to find all substrings of text T that are order isomorphic to pattern P when T(|T|=n) and P(|P|=m) are given. An O(mn+nqlogq+q!)-time algorithm using the O(m+q!) space for the order-preserving pattern matching problem has been proposed utilizing fingerprints of q-grams based on the factorial number system and the bad character heuristic. In this paper, we propose an O(mn+2q)-time algorithm using the O(m+2q) space for the order-preserving pattern matching problem, but utilizing fingerprints of q-grams converted to binary numbers. A comparative experiment using three types of time series data demonstrates that the proposed algorithm is faster than existing algorithms because it reduces the number of order isomorphism tests.

Order isomorphisms between bases of topologies

In this paper we will study the representations of isomorphisms between bases of topological spaces. It turns out that the perfect setting for this study is that of regular open subsets of complete metric spaces, but we have been able to show some results about arbitrary bases in complete metric spaces and also about regular open subsets in Hausdorff regular topological spaces.

Loewner’s theorem for maps on operator domains

AbstractThe classical Loewner’s theorem states that operator monotone functions on real intervals are described by holomorphic functions on the upper half-plane. We characterize local order isomorphisms on operator domains by biholomorphic automorphisms of the generalized upper half-plane, which is the collection of all operators with positive invertible imaginary part. We describe such maps in an explicit manner, and examine properties of maximal local order isomorphisms. Moreover, in the finite-dimensional case, we prove that every order embedding of a matrix domain is a homeomorphic order isomorphism onto another matrix domain.

Characterization of extension map on fuzzy weakly cut-stable map

<abstract><p>In this paper, based on a complete residuated lattice, we propose the definition of fuzzy weakly cut-stable map and prove the extension property of the fuzzy weakly cut-stable map. Following this, it is explored the conditions under which the extension map to be fuzzy order isomorphism.</p></abstract>

On endomorphisms of the inverse semigroup of convex order isomorphisms of a bounded rank which are generated by Rees congruences

On endomorphisms of the inverse semigroup of convex order isomorphisms of a bounded rank which are generated by Rees congruences

Spectral order isomorphisms and AW$AW$*‐factors

AbstractThe paper deals with spectral order isomorphisms in the framework of *‐algebras. We establish that every spectral order isomorphism between sets of all self‐adjoint elements (or between sets of all effects, or between sets of all positive elements) in *‐factors of Type I has a canonical form induced by a continuous function calculus and an isomorphism between projection lattices. In particular, this solves an open question about spectral order automorphisms of the set of all (bounded) self‐adjoint operators on an infinite‐dimensional Hilbert space. We also discuss spectral order isomorphisms preserving, in addition, orthogonality in both directions.

Dual spaces of operator systems

The aim of this article is to give an infinite dimensional analogue of a result of Choi and Effros concerning dual spaces of finite dimensional unital operator systems. An (not necessarily unital) operator system is a self-adjoint subspace of L(H), equipped with the induced matrix norm and the induced matrix cone. We say that an operator system T is dualizable if one can find an equivalent dual matrix norm on the dual space T⁎ such that under this dual matrix norm and the canonical dual matrix cone, T⁎ becomes a dual operator system. We show that an operator system T is dualizable if and only if the ordered normed space M∞(T)sa satisfies a form of bounded decomposition property. In this case,‖f‖d:=sup⁡{‖[fi,j(xk,l)]‖:x∈Mn(T)+;‖x‖≤1;n∈N}(f∈Mm(T⁎);m∈N), is the largest dual matrix norm that is equivalent to and dominated by the original dual matrix norm on T⁎ that turns it into a dual operator system, denoted by Td. It can be shown that Td is again dualizable. For every completely positive completely bounded map ϕ:S→T between dualizable operator systems, there is a unique weak-⁎-continuous completely positive completely bounded map ϕd:Td→Sd which is compatible with the dual map ϕ⁎. From this, we obtain a full and faithful functor from the category of dualizable operator systems to that of dualizable dual operator systems. Moreover, we will verify that if S is either a C⁎-algebra or a unital operator system, then S is dualizable and the canonical weak-⁎-homeomorphism from the unital operator system S⁎⁎ to the operator system (Sd)d is a completely isometric complete order isomorphism. Furthermore, via the duality functor above, the category of C⁎-algebras and that of unital operator systems (both equipped with completely positive complete contractions as their morphisms) can be regarded as full subcategories of the category of dual operator systems (with weak-⁎-continuous completely positive complete contractions as its morphisms). Consequently, a nice duality framework for operator systems is obtained, which includes all C⁎-algebras and all unital operator systems.