scholarly journals Generating sets of Reidemeister moves of oriented singular links and quandles

2018 ◽  
Vol 27 (14) ◽  
pp. 1850064 ◽  
Author(s):  
Khaled Bataineh ◽  
Mohamed Elhamdadi ◽  
Mustafa Hajij ◽  
William Youmans
Keyword(s):  
Algebraic Structure ◽  
Abelian Groups ◽  
Generating Set ◽  
Generating Sets ◽  
Reidemeister Moves ◽  
Singular Knots

We give a generating set of the generalized Reidemeister moves for oriented singular links. We then introduce an algebraic structure arising from the axiomatization of Reidemeister moves on oriented singular knots. We give some examples, including some non-isomorphic families of such structures over non-abelian groups. We show that the set of colorings of a singular knot by this new structure is an invariant of oriented singular knots and use it to distinguish some singular links.

scholarly journals Involutory biquandles and singular knots and links

2018 ◽  
Vol 16 (1) ◽  
pp. 469-489
Author(s):  
Khaled Bataineh ◽  
Hadeel Ghaith
Keyword(s):  
Algebraic Structure ◽  
Reidemeister Moves ◽  
Knots And Links ◽  
Singular Knots

AbstractWe define a new algebraic structure for singular knots and links. It extends the notion of a bikei (or involutory biquandle) from regular knots and links to singular knots and links. We call this structure a singbikei. This structure results from the generalized Reidemeister moves representing singular isotopy. We give several examples on singbikei and we use singbikei to distinguish several singular knots and links.

scholarly journals Minimal generating sets of directed oriented Reidemeister moves

2017 ◽  
Vol 26 (04) ◽  
pp. 1750016 ◽  
Author(s):  
Piotr Suwara
Keyword(s):  
Generating Set ◽  
Type Formula ◽  
Generating Sets ◽  
Reidemeister Moves ◽  
Different Types ◽  
Minimal Generating Set ◽  
Minimal Formula

Polyak proved that the set [Formula: see text] is a minimal generating set of oriented Reidemeister moves. One may distinguish between forward and backward moves, obtaining [Formula: see text] different types of moves, which we call directed oriented Reidemeister moves. In this paper, we prove that the set of eight directed Polyak moves [Formula: see text] is a minimal generating set of directed oriented Reidemeister moves. We also specialize the problem, introducing the notion of a [Formula: see text]-generating set for a link [Formula: see text]. The same set is proven to be a minimal [Formula: see text]-generating set for any link [Formula: see text] with at least two components. Finally, we discuss knot diagram invariants arising in the study of [Formula: see text]-generating sets for an arbitrary knot [Formula: see text], emphasizing the distinction between ascending and descending moves of type [Formula: see text].

scholarly journals GROWTH OF GENERATING SETS FOR DIRECT POWERS OF CLASSICAL ALGEBRAIC STRUCTURES

2010 ◽  
Vol 89 (1) ◽  
pp. 105-126 ◽  
Author(s):  
MARTYN QUICK ◽  
N. RUŠKUC
Keyword(s):  
General Theory ◽  
Algebraic Structure ◽  
Simple Structure ◽  
Module Algebra ◽  
Direct Power ◽  
Generating Set ◽  
Generating Sets ◽  
Infinite Case ◽  
Congruence Permutable Varieties ◽  
Constant Growth

AbstractFor an algebraic structure A denote byd(A) the smallest size of a generating set for A, and letd(A)=(d(A),d(A2),d(A3),…), where An denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequenced(A) when A is one of the classical structures—a group, ring, module, algebra or Lie algebra. We show that if A is finite thend(A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes thend(A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.

Additive bases via Fourier analysis

2021 ◽  
pp. 1-12
Author(s):  
Bodan Arsovski
Keyword(s):  
Abelian Group ◽  
Fourier Analysis ◽  
Vector Space ◽  
Abelian Groups ◽  
Finite Abelian Group ◽  
Probabilistic Method ◽  
Additive Basis ◽  
Generating Sets

Abstract Extending a result by Alon, Linial, and Meshulam to abelian groups, we prove that if G is a finite abelian group of exponent m and S is a sequence of elements of G such that any subsequence of S consisting of at least $$|S| - m\ln |G|$$ elements generates G, then S is an additive basis of G . We also prove that the additive span of any l generating sets of G contains a coset of a subgroup of size at least $$|G{|^{1 - c{ \in ^l}}}$$ for certain c=c(m) and $$ \in = \in (m) < 1$$ ; we use the probabilistic method to give sharper values of c(m) and $$ \in (m)$$ in the case when G is a vector space; and we give new proofs of related known results.

A Control System for a High-Quality Generating Set Equipped With a Two-Shaft Gas Turbine

1991 ◽  
Vol 113 (2) ◽  
pp. 290-295 ◽  
Author(s):  
H. Kumakura ◽  
T. Matsumura ◽  
E. Tsuruta ◽  
A. Watanabe
Keyword(s):  
Control System ◽  
Gas Turbine ◽  
Gas Turbines ◽  
High Quality ◽  
Constant Frequency ◽  
Generating Set ◽  
Generating Sets ◽  
Power Turbine ◽  
Computer Centers ◽  
Supply Systems

A control system has been developed for a high-quality generating set (150-kW) equipped with a two-shaft gas turbine featuring a variable power turbine nozzle. Because this generating set satisfies stringent frequency stability requirements, it can be employed as the direct electric power source for computer centers without using constant-voltage, constant-frequency power supply systems. Conventional generating sets of this kind have normally been powered by single-shaft gas turbines, which have a larger output shaft inertia than the two-shaft version. Good frequency characteristics have also been realized with the two-shaft gas turbine, which provides superior quick start ability and lower fuel consumption under partial loads.

scholarly journals Surface groups, infinite generating sets, and stable commutator length

2019 ◽  
Vol 150 (5) ◽  
pp. 2379-2386
Author(s):  
Dan Margalit ◽  
Andrew Putman
Keyword(s):  
Mapping Class Group ◽  
Class Group ◽  
Surface Group ◽  
Mapping Class ◽  
Surface Groups ◽  
Closed Curves ◽  
Generating Set ◽  
Generating Sets ◽  
Stable Commutator Length ◽  
Commutator Length

AbstractWe give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

scholarly journals Impartial achievement games for generating nilpotent groups

2019 ◽  
Vol 22 (3) ◽  
pp. 515-527
Author(s):  
Bret J. Benesh ◽  
Dana C. Ernst ◽  
Nándor Sieben
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Abelian Groups ◽  
Nilpotent Groups ◽  
Generating Set ◽  
Odd Order ◽  
Impartial Game ◽  
A Finite Group

AbstractWe study an impartial game introduced by Anderson and Harary. The game is played by two players who alternately choose previously-unselected elements of a finite group. The first player who builds a generating set from the jointly-selected elements wins. We determine the nim-numbers of this game for finite groups of the form{T\times H}, whereTis a 2-group andHis a group of odd order. This includes all nilpotent and hence abelian groups.

scholarly journals Computation of minimal homogeneous generating sets and minimal standard bases for ideals of free algebras

2018 ◽  
Vol 25 (3) ◽  
pp. 451-459
Author(s):  
Huishi Li
Keyword(s):  
Free Algebra ◽  
Standard Basis ◽  
Free Algebras ◽  
Grobner Basis ◽  
Finitely Generated ◽  
Generating Set ◽  
Standard Bases ◽  
Generating Sets ◽  
Monomial Ordering ◽  
Minimal Standard

AbstractLet {K\langle X\rangle=K\langle X_{1},\ldots,X_{n}\rangle} be the free algebra generated by {X=\{X_{1},\ldots,X_{n}\}} over a field K. It is shown that, with respect to any weighted {\mathbb{N}}-gradation attached to {K\langle X\rangle}, minimal homogeneous generating sets for finitely generated graded two-sided ideals of {K\langle X\rangle} can be algorithmically computed, and that if an ungraded two-sided ideal I of {K\langle X\rangle} has a finite Gröbner basis {{\mathcal{G}}} with respect to a graded monomial ordering on {K\langle X\rangle}, then a minimal standard basis for I can be computed via computing a minimal homogeneous generating set of the associated graded ideal {\langle\mathbf{LH}(I)\rangle}.

scholarly journals A GENERATING SET FOR THE AUTOMORPHISM GROUP OF A GRAPH PRODUCT OF ABELIAN GROUPS

2012 ◽  
Vol 22 (01) ◽  
pp. 1250003 ◽  
Author(s):  
L. J. CORREDOR ◽  
M. A. GUTIERREZ
Keyword(s):  
Automorphism Group ◽  
London Math ◽  
Abelian Groups ◽  
Graph Products ◽  
Graph Product ◽  
Finitely Generated ◽  
Generating Set ◽  
Labeled Graph

We find a set of generators for the automorphism group Aut G of a graph product G of finitely generated abelian groups entirely from a certain labeled graph. In addition, we find generators for the important subgroup Aut ⋆ G defined in [Automorphisms of graph products of abelian groups, to appear in Groups, Geometry and Dynamics]. We follow closely the plan of M. Laurence's paper [A generating set for the automorphism group of a graph group, J. London Math. Soc. (2)52(2) (1995) 318–334].

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