ON QUOTIENT STRUCTURE OF TAKASAKI QUANDLES

2013 ◽  
Vol 22 (12) ◽  
pp. 1341001 ◽  
Author(s):  
YONGJU BAE ◽  
SEONGJEONG KIM
Keyword(s):  
Abelian Group ◽  
Binary Operation ◽  
Identity Element ◽  
Algebraic Properties

A Takasaki quandle is defined by the binary operation a * b = 2b - a on an abelian group G. A Takasaki quandle depends on the algebraic properties of the underlying abelian group. In this paper, we will study the quotient structure of a Takasaki quandle in terms of its subquandle. If a subquandle X of a quandle Q is a subgroup of the underlying group Q, then we can define the quandle structure on the set {X * g | g ∈ Q}, which is called the quotient quandle of Q. Also we will study conditions for a subquandle X to be a subgroup of the underlying group when it contains the identity element.

On Anti-Commutative Algebras and Analytic Loops

1965 ◽  
Vol 17 ◽  
pp. 550-558 ◽  
Author(s):  
Arthur A. Sagle
Keyword(s):  
Lie Algebra ◽  
Lie Group ◽  
Binary Operation ◽  
Identity Element ◽  
Unique Element ◽  
Commutative Algebras ◽  
Algebraic Properties

In (4) Malcev generalizes the notion of the Lie algebra of a Lie group to that of an anti-commutative "tangent algebra" of an analytic loop. In this paper we shall discuss these concepts briefly and modify them to the situation where the cancellation laws in the loop are replaced by a unique two-sided inverse. Thus we shall have a set H with a binary operation xy defined on it having the algebraic properties(1.1) H contains a two-sided identity element e;(1.2) for every x ∊ H, there exists a unique element x-1 ∊ H such that xx-1 = x-1x = e;

GENERALIZED COMPOSITIONS OF FUZZY RELATIONS

2002 ◽  
Vol 10 (supp01) ◽  
pp. 149-163 ◽  
Author(s):  
JÓZEF DREWNIAK ◽  
KRZYSZTOF KULA
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Fuzzy Relations ◽  
Triangular Norms ◽  
Algebraic Properties

We examine compositions of fuzzy relations based on a binary operation *. We discuss the dependences between algebraic properties of the operation * and the induced sup –* composition. It is examined independently for monotone operations, for operations with idempotent, zero or identity element, for distributive and associative operations. Finally, we present consequences of these results for compositions based on triangular norms, triangular conorms and uninorms.

scholarly journals Sequential completeness of quotient groups

2000 ◽  
Vol 61 (1) ◽  
pp. 129-150 ◽  
Author(s):  
Dikran Dikranjan ◽  
Michael Tkačenko
Keyword(s):  
Abelian Group ◽  
Direct Products ◽  
Topological Groups ◽  
Complete Group ◽  
Sequential Completeness ◽  
Algebraic Properties ◽  
Countable Compactness ◽  
Quotient Groups

We discuss various generalisations of countable compactness for topological groups that are related to completeness. The sequentially complete groups form a class closed with respect to taking direct products and closed subgroups. Surprisingly, the stronger version of sequential completeness called sequential h-completeness (all continuous homomorphic images are sequentially complete) implies pseudocompactness in the presence of good algebraic properties such as nilpotency. We also study quotients of sequentially complete groups and find several classes of sequentially q-complete groups (all quotients are sequentially complete). Finally, we show that the pseudocompact sequentially complete groups are far from being sequentially q-complete in the following sense: every pseudocompact Abelian group is a quotient of a pseudocompact Abelian sequentially complete group.

Quasi-linear Cycle Sets and the Retraction Problem for Set-theoretic Solutions of the Quantum Yang-Baxter Equation

2016 ◽  
Vol 23 (01) ◽  
pp. 149-166 ◽  
Author(s):  
Wolfgang Rump
Keyword(s):  
Abelian Group ◽  
Algebraic System ◽  
Binary Operation ◽  
Group Structure ◽  
Baxter Equation ◽  
Proper Extension ◽  
Group Structures ◽  
New Evidence

Cycle sets were introduced to reduce non-degenerate unitary Yang-Baxter maps to an algebraic system with a single binary operation. Every finite cycle set extends uniquely to a finite cycle set with a compatible abelian group structure. Etingof et al. introduced affine Yang-Baxter maps. These are equivalent to cycle sets with a specific abelian group structure. Abelian group structures have also been essential to get partial results for the still unsolved retraction problem. We introduce two new classes of cycle sets with an underlying abelian group structure and show that they can be transformed into each other while keeping the group structure fixed. This leads to a proper extension of the retractibility conjecture and new evidence for its truth.

Exceptional algebraic properties of the three quadratic irrationalities observed in quasicrystals

2001 ◽  
Vol 79 (2-3) ◽  
pp. 687-696 ◽  
Author(s):  
Z Masáková ◽  
J Patera ◽  
E Pelantová
Keyword(s):  
Binary Operation ◽  
Rational Numbers ◽  
Point Sets ◽  
Irrational Numbers ◽  
Algebraic Properties ◽  
Equivalent Characterization

There are only three irrationalities directly related to experimentally observed quasicrystals, namely, those which appear in extensions of rational numbers by Ö5, Ö2, Ö3. In this article, we demonstrate that the algebraically defined aperiodic point sets with precisely these three irrational numbers play an exceptional role. The exceptional role stems from the possibility of equivalent characterization of these point sets using one binary operation. PACS Nos.: 61.90+d, 61.50-f

Set Theory Generated by Abelian Group Theory

1997 ◽  
Vol 3 (1) ◽  
pp. 1-16 ◽  
Author(s):  
Paul C. Eklof
Keyword(s):  
Group Theory ◽  
Abelian Group ◽  
Set Theory ◽  
Identity Element ◽  
Historical Context ◽  
Group Operation ◽  
Independence Result ◽  
Comprehensive Survey ◽  
New Ideas ◽  
Free Abelian Groups

Introduction. This survey is intended to introduce to logicians some notions, methods and theorems in set theory which arose—largely through the work of Saharon Shelah—out of (successful) attempts to solve problems in abelian group theory, principally the Whitehead problem and the closely related problem of the existence of almost free abelian groups. While Shelah's first independence result regarding the Whitehead problem used established set-theoretical methods (discussed below), his later work required new ideas; it is on these that we focus. We emphasize the nature of the new ideas and the historical context in which they arose, and we do not attempt to give precise technical definitions in all cases, nor to include a comprehensive survey of the algebraic results.In fact, very little algebraic background is needed beyond the definitions of group and group homomorphism. Unless otherwise specified, we will use the word “group” to refer to an abelian group, that is, the group operation is commutative. The group operation will be denoted by +, the identity element by 0, and the inverse of a by −a. We shall use na as an abbreviation for a + a + … + a [n times] if n is positive, and na = (−n)(−a) if n is negative.

scholarly journals Extending abelian groups to rings

2007 ◽  
Vol 82 (3) ◽  
pp. 297-314 ◽  
Author(s):  
Lynn M. Batten ◽  
Robert S. Coulter ◽  
Marie Henderson
Keyword(s):  
Abelian Group ◽  
Commutative Ring ◽  
Equivalence Relation ◽  
Binary Operation ◽  
Sufficient Conditions ◽  
Additive Group ◽  
Necessary And Sufficient Conditions ◽  
Odd Order ◽  
Weak Isomorphism ◽  
Necessary And Sufficient

AbstractFor any abelian group G and any function f: G → G we define a commutative binary operation or ‘multiplication’ on G in terms of f. We give necessary and sufficient conditions on f for G to extend to a commutative ring with the new multiplication. In the case where G is an elementary abelian p–group of odd order, we classify those functions which extend G to a ring and show, under an equivalence relation we call weak isomorphism, that there are precisely six distinct classes of rings constructed using this method with additive group the elementary abelian p–group of odd order p2.

A characteristic property of elementary 2-groups of rank six

2011 ◽  
Vol 48 (3) ◽  
pp. 354-370
Author(s):  
Sándor Szabó
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Characteristic Property ◽  
Identity Element ◽  
Finite Abelian Group

Consider a finite abelian group G which is a direct product of its subsets A and B both containing the identity element e. If the non-periodicity of A and B forces that neither A nor B can span the whole G, then G must be an elementary 2-group of rank six.

Expansions of dense linear orders with the intermediate value property

2001 ◽  
Vol 66 (4) ◽  
pp. 1783-1790 ◽  
Author(s):  
Chris Miller
Keyword(s):  
Binary Operation ◽  
Identity Element ◽  
Linear Orders ◽  
Ordered Group ◽  
Definable Sets ◽  
Noncyclic Subgroup ◽  
Preceding Paragraph ◽  
Intermediate Value ◽  
Familiar Result ◽  
Definable Function

Let ℜ be an expansion of a dense linear order (R, <) without endpoints having the intermediate value property, that is, for all a, b ∈ R, every continuous (parametrically) definable function f: [a, b] → R takes on all values in R between f(a) and f(b). Every expansion of the real line (ℝ, <), as well as every o-minimal expansion of (R, <), has the intermediate value property. Conversely, some nice properties, often associated with expansions of (ℝ, <) or with o-minimal structures, hold for sets and functions definable in ℜ. For example, images of closed bounded definable sets under continuous definable maps are closed and bounded (Proposition 1.10).Of particular interest is the case that ℜ expands an ordered group, that is, ℜ defines a binary operation * such that (R, <, *) is an ordered group. Then (R, *) is abelian and divisible (Proposition 2.2). Continuous nontrivial definable endo-morphisms of (R, *) are surjective and strictly monotone, and monotone nontrivial definable endomorphisms of (R, *) are strictly monotone, continuous and surjective (Proposition 2.4). There is a generalization of the familiar result that every proper noncyclic subgroup of (ℝ, +) is dense and codense in ℝ: If G is a proper nontrivial subgroup of (R, *) definable in ℜ, then either G is dense and codense in R, or G contains an element u such that (R, <, *, e, u, G) is elementarily equivalent to (ℚ, <, +, 0, 1, ℤ), where e denotes the identity element of (R, *) (Theorem 2.3).Here is an outline of this paper. First, we deal with some basic topological results. We then assume that ℜ expands an ordered group and establish the results mentioned in the preceding paragraph. Some examples are then given, followed by a brief discussion of analytic results and possible limitations. In an appendix, an explicit axiomatization (used in the proof of Theorem 2.3) is given for the complete theory of the structure (ℚ, <, +, 0, 1, ℤ).

On quotient structure of Takasaki quandles II

2014 ◽  
Vol 23 (07) ◽  
pp. 1460012 ◽  
Author(s):  
Yongju Bae ◽  
Seongjeong Kim
Keyword(s):  
Abelian Group ◽  
Knot Theory ◽  
Binary Operation ◽  
The Other ◽  
Other Hand

A Takasaki quandle (T(G), *) is a quandle under the binary operation * defined by a*b = 2b-a for an abelian group (G, +). In this paper, we will show that if a subquandle X of a Takasaki quandle G is a image of subgroup of G under a quandle automorphism of T(G), then the set {X * g | g ∈ G} is a quandle under the binary operation *′ defined by (X * g) *′ (X * h) = X * (g * h). On the other hand, the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] can be applied to the Takasaki quandles. In this paper, we will review the quotient structure studied in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156], and show that the quotient quandle coincides with the quotient quandle defined by Bunch, Lofgren, Rapp and Yetter in [On quotients of quandles, J. Knot Theory Ramifications 19(9) (2010) 1145–1156] for connected Takasaki quandles.

Sign in / Sign up

Export Citation Format

Share Document