Set-theoretic solutions to the Yang–Baxter equation, skew-braces, and related near-rings

2019 ◽  
Vol 18 (08) ◽  
pp. 1950145
Author(s):  
Wolfgang Rump
Keyword(s):  
Residue Field ◽  
Ring Structure ◽  
Baxter Equation ◽  
Exponential Map ◽  
Exponential Form ◽  
Degenerate Solution ◽  
Terminal Object ◽  
Theoretic Solution ◽  
Initial Object ◽  
Central Residue

Skew-braces have been introduced recently by Guarnieri and Vendramin. The structure group of a non-degenerate solution to the Yang–Baxter equation is a skew-brace, and every skew-brace gives a set-theoretic solution to the Yang–Baxter equation. It is proved that skew-braces arise from near-rings with a distinguished exponential map. For a fixed skew-brace, the corresponding near-rings with exponential form a category. The terminal object is a near-ring of self-maps, while the initial object is a near-ring which gives a complete invariant of the skew-brace. The radicals of split local near-rings with a central residue field [Formula: see text] are characterized as [Formula: see text]-braces with a compatible near-ring structure. Under this correspondence, [Formula: see text]-braces are radicals of local near-rings with radical square zero.

scholarly journals Left semi-braces and solutions of the Yang–Baxter equation

2019 ◽  
Vol 31 (1) ◽  
pp. 241-263 ◽  
Author(s):  
Eric Jespers ◽  
Arne Van Antwerpen
Keyword(s):  
Positive Integer ◽  
Polynomial Identity ◽  
Baxter Equation ◽  
Polynomial Algebras ◽  
Finite Set ◽  
Theoretic Solution ◽  
Kirillov Dimension ◽  
Do So

Abstract Let {r\colon X^{2}\rightarrow X^{2}} be a set-theoretic solution of the Yang–Baxter equation on a finite set X. It was proven by Gateva-Ivanova and Van den Bergh that if r is non-degenerate and involutive, then the algebra {K\langle x\in X\mid xy=uv\text{ if }r(x,y)=(u,v)\rangle} shares many properties with commutative polynomial algebras in finitely many variables; in particular, this algebra is Noetherian, satisfies a polynomial identity and has Gelfand–Kirillov dimension a positive integer. Lebed and Vendramin recently extended this result to arbitrary non-degenerate bijective solutions. Such solutions are naturally associated to finite skew left braces. In this paper we will prove an analogue result for arbitrary solutions {r_{B}} that are associated to a left semi-brace B; such solutions can be degenerate or can even be idempotent. In order to do so, we first describe such semi-braces and then prove some decompositions results extending those of Catino, Colazzo and Stefanelli.

Hölder categories

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
Anthony Hager ◽  
Jorge Martínez
Keyword(s):  
Order Unit ◽  
Real Numbers ◽  
Initial Element ◽  
Ordered Groups ◽  
Lattice Ordered Groups ◽  
Terminal Object ◽  
Epireflective Hull ◽  
Strong Order ◽  
Archimedean Lattice ◽  
Initial Object

AbstractHölder categories are invented to provide an axiomatic foundation for the study of categories of archimedean lattice-ordered algebraic structures. The basis of such a study is Hölder’s Theorem (1908), stating that the archimedean totally ordered groups are precisely the subgroups of the additive real numbers ℝ with the usual addition and ordering, which remains the single most consequential result in the studies of lattice-ordered algebraic systems since Birkhoff and Fuchs to the present.This study originated with interest in W*, the category of all archimedean lattice-ordered groups with a designated strong order unit, and the ℓ-homomorphisms which preserve those units, and, more precisely, with interest in the epireflections on W*. In the course of this study, certain abstract notions jumped to the forefront. Two of these, in particular, seem to have been mostly overlooked; some notion of simplicity appears to be essential to any kind of categorical study of W*, as are the quasi-initial objects in a category. Once these two notions have been brought into the conversation, a Hölder category may then be defined as one which is complete, well powered, and in which(a) the initial object I is simple, and(b) there is a simple quasi-initial coseparator R.In this framework it is shown that the epireflective hull of R is the least monoreflective class. And, when I = R — that is, the initial element is simple and a coseparator — a theorem of Bezhanishvili, Morandi, and Olberding, for bounded archimedean f-algebras with identity, can be be generalized, as follows: for any Hölder category subject to the stipulation that the initial object is a simple coseparator, every uniformly nontrivial reflection — meaning that the reflection of each non-terminal object is non-terminal — is a monoreflection.Also shown here is the fact that the atoms in the class of epireflective classes are the epireflective hulls of the simple quasi-initial objects. From this observation one easily deduces a converse to the result of Bezhanishvili, Morandi, and Olberding: if in a Hölder category every epireflection is a monoreflection, then the initial object is a coseparator.

On the Limitations of Sketches

1992 ◽  
Vol 35 (3) ◽  
pp. 287-294
Author(s):  
Michael Barr ◽  
Charles Wells
Keyword(s):  
Higher Order ◽  
Terminal Object ◽  
Initial Object

AbstractCall a category "sketchable" if it is the category of models in sets of some sketch. This paper explores the subtle boundary between sketchable and non-sketchable categories. We show that the category of small categories that have at least one initial object and functors that take an initial object to an initial object is sketchable. The same is true for weak initial objects, but is false for subinitial objects (that every object has at most one arrow to). Analogous results hold if we substitute finite limits for terminal object. We also show that the category of groups and center-preserving homomorphisms is not sketchable. We describe briefly how "higher-order" sketches can fill these gaps.

Integer-valued skew polynomials

2020 ◽  
pp. 2150114
Author(s):  
Nicholas J. Werner
Keyword(s):  
Valuation Ring ◽  
Integral Domain ◽  
Residue Field ◽  
Discrete Valuation Ring ◽  
Ring Structure ◽  
Polynomial Rings ◽  
Skew Polynomial Ring ◽  
Skew Polynomials ◽  
Skew Polynomial Rings ◽  
Skew Polynomial

For a commutative integral domain [Formula: see text] with field of fractions [Formula: see text], the ring of integer-valued polynomials on [Formula: see text] is [Formula: see text]. In this paper, we extend this construction to skew polynomial rings. Given an automorphism [Formula: see text] of [Formula: see text], the skew polynomial ring [Formula: see text] consists of polynomials with coefficients from [Formula: see text], and with multiplication given by [Formula: see text] for all [Formula: see text]. We define [Formula: see text], which is the set of integer-valued skew polynomials on [Formula: see text]. When [Formula: see text] is not the identity, [Formula: see text] is noncommutative and evaluation behaves differently than it does for ordinary polynomials. Nevertheless, we are able to prove that [Formula: see text] has a ring structure in many cases. We show how to produce elements of [Formula: see text] and investigate its properties regarding localization and Noetherian conditions. Particular attention is paid to the case where [Formula: see text] is a discrete valuation ring with finite residue field.

Small Angle Electron Scattering From Vacuum Condensed Metallic Films

1968 ◽  
Vol 26 ◽  
pp. 380-381
Author(s):  
J. Silcox ◽  
R. H. Wade
Keyword(s):  
Electron Scattering ◽  
Small Angle ◽  
Ring Structure ◽  
Two Dimensional ◽  
Metallic Films ◽  
Micro Structure ◽  
Angle Scattering ◽  
The Fourier Transform ◽  
Source Of Information ◽  
Kinematical Theory

Recent work has drawn attention to the possibilities that small angle electron scattering offers as a source of information about the micro-structure of vacuum condensed films. In particular, this serves as a good detector of discontinuities within the films. A review of a kinematical theory describing the small angle scattering from a thin film composed of discrete particles packed close together will be presented. Such a model could be represented by a set of cylinders packed side by side in a two dimensional fluid-like array, the axis of the cylinders being normal to the film and the length of the cylinders becoming the thickness of the film. The Fourier transform of such an array can be regarded as a ring structure around the central beam in the plane of the film with the usual thickness transform in a direction normal to the film. The intensity profile across the ring structure is related to the radial distribution function of the spacing between cylinders.

Immunoelectron microscope detection of C-type natriuretic peptide in normal human atrial tissue

1993 ◽  
Vol 51 ◽  
pp. 388-389
Author(s):  
Chi-Ming Wei ◽  
Margaret Hukee ◽  
Christopher G.A. McGregor ◽  
John C. Burnett
Keyword(s):  
Amino Acid ◽  
Natriuretic Peptide ◽  
Vascular Tone ◽  
Cardiac Tissue ◽  
Ring Structure ◽  
Terminal Amino Acid ◽  
Amino Acid Peptide ◽  
Normal Human ◽  
Terminal Amino ◽  
The Brain

C-type natriuretic peptide (CNP) is a newly identified peptide that is structurally related to atrial (ANP) and brain natriuretic peptide (BNP). CNP exists as a 22-amino acid peptide and like ANP and BNP has a 17-amino acid ring formed by a disulfide bond. Unlike these two previously identified cardiac peptides, CNP lacks the COOH-terminal amino acid extension from the ring structure. ANP, BNP and CNP decrease cardiac preload, but unlike ANP and BNP, CNP is not natriuretic. While ANP and BNP have been localized to the heart, recent investigations have failed to detect CNP mRNA in the myocardium although small concentrations of CNP are detectable in the porcine myocardium. While originally localized to the brain, recent investigations have localized CNP to endothelial cells consistent with a paracrine role for CNP in the control of vascular tone. While CNP has been detected in cardiac tissue by radioimmunoassay, no studies have demonstrated CNP localization in normal human heart by immunoelectron microscopy.

Sign in / Sign up

Export Citation Format

Share Document