Tensor products and regularity properties of Cuntz semigroups

The Cuntz semigroup of a C*-algebra is an important invariant in the structure and classification theory of C*-algebras. It captures more information than K-theory but is often more delicate to handle. We systematically study the lattice and category theoretic aspects of Cuntz semigroups. Given a C*-algebra $A$, its (concrete) Cuntz semigroup $Cu(A)$ is an object in the category $Cu$ of (abstract) Cuntz semigroups, as introduced by Coward, Elliott and Ivanescu. To clarify the distinction between concrete and abstract Cuntz semigroups, we will call the latter $Cu$-semigroups. We establish the existence of tensor products in the category $Cu$ and study the basic properties of this construction. We show that $Cu$ is a symmetric, monoidal category and relate $Cu(A\otimes B)$ with $Cu(A)\otimes_{Cu}Cu(B)$ for certain classes of C*-algebras. As a main tool for our approach we introduce the category $W$ of pre-completed Cuntz semigroups. We show that $Cu$ is a full, reflective subcategory of $W$. One can then easily deduce properties of $Cu$ from respective properties of $W$, e.g. the existence of tensor products and inductive limits. The advantage is that constructions in $W$ are much easier since the objects are purely algebraic. We also develop a theory of $Cu$-semirings and their semimodules. The Cuntz semigroup of a strongly self-absorbing C*-algebra has a natural product giving it the structure of a $Cu$-semiring. We give explicit characterizations of $Cu$-semimodules over such $Cu$-semirings. For instance, we show that a $Cu$-semigroup $S$ tensorially absorbs the $Cu$-semiring of the Jiang-Su algebra if and only if $S$ is almost unperforated and almost divisible, thus establishing a semigroup version of the Toms-Winter conjecture.Comment: 195 pages; revised version; several proofs streamlined; some results corrected, in particular added 5.2.3-5.2.

Hyper-dependence, hyper-ageing properties and analogies between them: a semigroup-based approach

In previous papers, evolution of dependence and ageing, for vectors of non-negative random variables, have been separately considered. Some analogies between the two evolutions emerge however in those studies. In the present paper, we propose a unified approach, based on semigroup arguments, explaining the origin of such analogies and relations among properties of stochastic dependence and ageing

Finite Computational Structures and Implementations

What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only partial answers to these questions. In order to make these problems more precise, we describe an abstract algebraic definition of classical computation, generalizing traditional models to semigroups. The mathematical abstraction also allows the investigation of different computing paradigms (e.g. cellular automata, reversible computing) in the same framework. Here we summarize the main questions and recent results of the research of finite computation.Comment: 12 pages, 3 figures, will be presented at CANDAR'16 and final version published by IEEE Computer Societ

RIGHT PURE UNI-SOFT IDEALS OF ORDERED SEMIGROUPS

In this paper, we initiate the study of pure uni-soft ideals in ordered semigroups. The soft version of right pure ideals in ordered semigroups is considered which is an extension of the concept of right pure ideal in ordered semigroups. We also give the main result for right pure uni-soft ideals in ordered semigroups and characterize right weakly regular ordered semigroups in terms of right pure uni-soft ideals

Abstract bivariant Cuntz semigroups

We show that abstract Cuntz semigroups form a closed symmetric monoidal category. Thus, given Cuntz semigroups S and T, there is another Cuntz semigroup JS, TK playing the role of morphisms from S to T. Applied to C*-algebras A and B, the semigroup JCu(A),Cu(B)K should be considered as the target in analogues of the UCT for bivariant theories of Cuntz semigroups. Abstract bivariant Cuntz semigroups are computable in a number of interesting cases. We explore its behaviour under the tensor product with the Cuntz semigroup of strongly self-absorbing C*-algebras and the Jacelon-Razak algebra. We also show that order-zero maps between C*-algebras naturally define elements in the respective bivariant Cuntz semigroup

History and new possible research directions of hyperstructures

We present a summary of the origins and current developments of the theory of algebraic hyperstructures. We also sketch some possible lines of research

A new classification of hemirings through double-framed soft h-ideals

Due to lack of parameterization, various ordinary uncertainty theories like theory of fuzzy sets, and theory of probability cannot solve complicated problems of economics and engineering involving uncertainties. The aim of the present paper was to provide an appropriate mathematical tool for solving such type of complicated problems. For the said purpose, the notion of double-framed soft sets in hemirings is introduced. As h-ideals of hemirings play a central role in the structural theory, therefore, we developed a new type of subsystem of hemirings. Double-framed soft left (right) h-ideal, double-framed soft h-bi-ideals and double-framed soft h-quasi-ideals of hemiring are determined. These concepts are elaborated through suitable examples. Furthermore, we are bridging ordinary h-ideals and double-framed soft h-ideals of hemirings through double-framed soft including sets and characteristic double-framed soft functions. It is also shown that every double-framed soft h-quasi-ideal is double-framed soft h-bi-ideal but the converse inclusion does not hold. A well-known class of hemrings i.e. h-hemiregular hemirings is characterized by the properties of these newly developed double-framed soft h-ideals o