Homotopy and semisymmetry of quasigroups

The study of quasigroup homotopies reduces to the study of homomorphisms between semisymmetric quasigroups. In particular, the study of homotopies between central quasigroups reduces to the study of homomorphisms between entropic semisymmetric quasigroups. Received December 20, 1996; accepted in final form September 17, 1997.     

Modular posets and semigroups

Posets and poset homomorphisms (preserving both order and parallelism) have been shown to form a category which is equivalent to the category of pogroupoids and their homomorphisms. Among the posets those posets whose associated pogroupoids are semigroups are identified as being precisely those posets which are (C ₂+1)-free. In the case of lattices this condition means that the lattice is alsoN ₅-free and hence modular. Using the standard connection: semigroup to poset to pogroupoid, it is observed that in many cases the image pogroupoid obtained is a semigroup even if quite different from the original one. The nature of this mapping appears intriguing in the poset setting and may well be so seen from the semigroup theory viewpoint.     

On the functoriality of cohomology of categories

In this paper we show that the Baues-Wirsching complex used to define cohomology of categories is a 2-functor from a certain 2-category of natural systems of abelian groups to the 2-category of chain complexes, chain homomorphisms and relative homotopy classes of chain homotopies. As a consequence we derive (co)localization theorems for this cohomology.     

On linear representations of quasigroups

For homotopies of quasigroups, an analog of the fundamental theorem on homomorphisms does not hold in general. In this paper, we consider two approaches that allow one to obtain an analog of this theorem: the introduction of strict homotopies and the move from quasigroups to three-sorted quasigroups.     

Computing all solutions to polynomial systems using homotopy continuation

In a previous paper we described a new method for defining homotopies for finding all solutions to polynomial systems. A major feature of this new approach is that the start system for the homotopy need not be a “random” or “generic” system. Also, homotopy paths are strictly increasing in the homotopy parameter. In this paper we establish some principles of implementation and report on the performance of programs that use the new homotopies. A feature of our implementation is that we eliminate divergent paths entirely. We include performance statistics for homotopies derived from more traditional approaches for comparison. Generally, the new approach is faster and more reliable.     

P-systems in regular semigroups

In this paper, firstly it is shown that a regular semigroup S becomes a regular *-semigroup (in the sense of [1]) if and only if S has a certain subset called a p-system. Secondly, all the normal *-bands are completely described in terms of rectangular *-bands (square bands) and transitive systems of homomorphisms of rectangular *-bands. Further, it is shown that an orthodox semigroup S becomes a regular *-semigroup if there is a p-system F of the band E_S of idempotents of S such that F∋e, E_S∋t, e≥t imply t∈F. By using this result, it is also shown that F is a p-system of a generalized inverse semigroup S if and only if F is a p-system of F_S. Dedicated to Professor L. M. Gluskin on his 60th birthday     

Clifford semigroups as functors and their cohomology

We prove that the category of Clifford semigroups and prehomomorphisms CSP\mathcal{CSP} is isomorphic to a certain subcategory of the category of diagrams over groups. Under this isomorphism, Clifford semigroups are identified with certain functors. As an application of the isomorphism theorem, we show that the category with objects commutative inverse semigroups having the same semilattice of idempotents and with morphisms, the inverse semigroup homomorphisms that fix the semilattice, imbeds into a category of right modules over a certain ring. Also we find a very close relationship between the cohomology groups of a commutative inverse monoid and the cohomology groups of the colimit group of the functor giving the monoid.     

Data compression with homomorphism in covering information systems

In reality we are always faced with a large number of complex massive databases. In this work we introduce the notion of a homomorphism as a kind of tool to study data compression in covering information systems. The concepts of consistent functions related to covers are first defined. Then, by classical extension principle the concepts of covering mapping and inverse covering mapping are introduced and their properties are studied. Finally, the notions of homomorphisms of information systems based on covers are proposed, and it is proved that a complex massive covering information system can be compressed into a relatively small-scale information system and its attribute reduction is invariant under the condition of homomorphism, that is, attribute reductions in the original system and image system are equivalent to each other.     

Fundamental group in o-minimal structures with definable Skolem functions

In this paper we work in an arbitrary o-minimal structure with definable Skolem functions and prove that definably connected, locally definable manifolds are uniformly definably path connected, have an admissible cover by definably simply connected, open definable subsets and, definable paths and definable homotopies on such locally definable manifolds can be lifted to locally definable covering maps. These properties allow us to obtain the main properties of the general o-minimal fundamental group, including: invariance and comparison results; existence of universal locally definable covering maps; monodromy equivalence for locally constant o-minimal sheaves – from which one obtains, as in algebraic topology, classification results for locally definable covering maps, o-minimal Hurewicz and Seifert–van Kampen theorems.     

A note on representation of pseudovariant maps

We define and study the notions of pseudovariant maps and pseudogeny maps on G-spaces. We prove that the set of all pseudogenies of a locally comapct G-space is a group. We further obtain representation of pseudovariant maps in terms of pseudogenies. Finally, we obtain Tietze type extension result for pseudovariant homotopies defined on locally compact second countable G-spaces.     

Homomorphic actions of ℍ on a compact groupon a compact group

The way the additive semigroup [0, ∞) carrying the usual topology can act as a semigroup of homomorphisms on a compact group is studied in this paper. It is shown that such actions of [0, ∞) on a compact group G can be completely described in terms of continuous homomorphisms from [0, ∞) into G. Furthermore, it is proven that each such action of [0, ∞) on a compact group G has a unique extension to an action of the additive group of all real numbers endowed with the usual topology on G so that a similar characterization is also valid for actions of the real numbers on a compact group.     

The connecting homomorphism for <Emphasis Type="Italic">K</Emphasis>-theory of generalized free products

We interpret the connecting homomorphism of the long exact sequence of algebraic K-groups associated to a category with cofibrations and two notions of weak equivalence. One notion of weak equivalence is finer than the other and certain technical conditions involving mapping cylinders must also be satisfied. Such situations arise when one considers certain free product diagrams in the category of rings, such as those that arise in applications of the Seifert-van Kampen theorem where all fundamental group homomorphisms are injective. The techniques follow Waldhausen’s approach to algebraic K-theory of categories with cofibrations and weak equivalences.     

非线性Lipschitz算子半群的表示

本文通过引入若干Lipschitz对偶概念,将非线性Lipschitz算子半群对偶映射到Lipschitz对偶空间中,使其转化为线性算子半群。该线性算子半群被证明是一个C_0~*-半群,因而是某个C_0-半群的对偶半群。从而证明了,在等距意义下,一个非线性Lipschitz算子半群可以延拓为一个C_0-半群。基于这些结论,本文给出了一系列全新的非线性Lipschitz算子半群的表示公式。     

Various notions of weak injectivity over Clifford semigroups

We study various notions of weak injectivity of acts over a Clifford semigroup. In addition we characterize weak self-injectivities of several classes of Clifford semigroups constructed via particular kinds of groups and group homomorphisms. Research of X. Zhang supported by the China Scholarship Council No. 2006101056. Research of Y. Wang supported by National Natural Science Foundation of China Research Grant 10571181.     

两类完全正则半群的同态和同余

本文研究了密群的同态和纯正群的幂等元分离同余.定义了密群的半格类之间的一种映射,利用该映射,半格的同态及完全单半群上的同态刻画了密群间的同态.利用群核正规系讨论了纯正群的幂等元分离同余.并给出其一种简单的表示形式.     

Positive reynolds operators and generating derivations

It is shown that the spectrum of a positive Reynolds operator on C₀(X) is contained in the disc centered at 1/2 with radius 1/2. Moreover, every positive Reynolds operator T with dense range is injective. In this case, the operator D = 1 — T^?1 is a densely defined derivation, which generates a one — parameter semigroup of algebra homomorphisms. This semigroup yields an integral representation of T. Along the way, it is proved that a densely defined closed derivation D generates a semigroup if, and only if, R(1, D) exists and is a positive operator.     

Basic study of ℜ-semigroups and their homomorphisms

In this paper we study ?-semigroups from the standpoint of their homomorphisms into the semigroup IR of positive real numbers under addition. Every ?⁺-semigroup S is isomorphic with a subdirect product of a structure group G and a semigroup of positive real numbers. It is determined by G and a positive real valued function cp on G. We also study the semigroup Hom(S,T) of homomorphisms of an ?-semigroup S into another ?-semigroup T. Each element of Hom(S,T) can be described in terms of an element of Hom(S,IR₊) and an element of Hom(S,H) where H is a structure group of T.     

Completeness and an extension theorem for continuous fuzzy linear order-homomorphisms

In this paper, the notions of λ-Cauchy net and completeness in L-topological vector spaces are introduced, and some of their properties are studied. An extension theorem for continuous fuzzy linear order homomorphisms is proved.     

A Realization of Hyperrings

The purpose of this article is to present certain results arising from a study of theory of hyperrings. By a hyperring we mean a Krasner hyperring, that is, a triple (R, +,·) is such that (R, +) is a canonical hypergroup, (R, ·) is a semigroup with a zero 0 where 0 is the scalar identity of (R, +) and · is distributive over + . In this article, we define the notions of normal, prime, maximal, and Jacobson radical of a hyperring and by considering these notions we obtain some results. We define hyperring of fractions and hyper-valuation on a hyperring. For this, as in the classical case, we need a mapping from R onto an ordered group G. Finally, we shall state and prove the Chinese Remainder Theorem for the case of hyperrings.     

Remarks on Priestley duality for distributive lattices

The notion of a Priestley relation between Priestley spaces is introduced, and it is shown that there is a duality between the category of bounded distributive lattices and 0-preserving join-homomorphisms and the category of Priestley spaces and Priestley relations. When restricted to the category of bounded distributive lattices and 0-1-preserving homomorphisms, this duality yields essentially Priestley duality, and when restricted to the subcategory of Boolean algebras and 0-preserving join-homomorphisms, it coincides with the Halmos-Wright duality. It is also established a duality between 0-1-sublattices of a bounded distributive lattice and certain preorder relations on its Priestley space, which are called lattice preorders. This duality is a natural generalization of the Boolean case, and is strongly related to one considered by M. E. Adams. Connections between both kinds of dualities are studied, obtaining dualities for closure operators and quantifiers. Some results on the existence of homomorphisms lying between meet and join homomorphisms are given in the Appendix.