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.

scholarly journals Gelfand-Kirillov Dimensions of Modules over Differential Difference Algebras

2016 ◽  
Vol 23 (04) ◽  
pp. 701-720 ◽  
Author(s):  
Xiangui Zhao ◽  
Yang Zhang
Keyword(s):  
Computational Method ◽  
Finitely Generated ◽  
Finitely Generated Module ◽  
Polynomial Algebras ◽  
Ore Extensions ◽  
Basis Method ◽  
Finitely Generated Modules ◽  
Kirillov Dimension

Differential difference algebras are generalizations of polynomial algebras, quantum planes, and Ore extensions of automorphism type and of derivation type. In this paper, we investigate the Gelfand-Kirillov dimension of a finitely generated module over a differential difference algebra through a computational method: Gröbner-Shirshov basis method. We develop the Gröbner-Shirshov basis theory of differential difference algebras, and of finitely generated modules over differential difference algebras, respectively. Then, via Gröbner-Shirshov bases, we give algorithms for computing the Gelfand-Kirillov dimensions of cyclic modules and finitely generated modules over differential difference algebras.

Type A Weyl Group Multiple Dirichlet Series

2011 ◽  
Author(s):  
Ben Brubaker ◽  
Daniel Bump ◽  
Solomon Friedberg
Keyword(s):  
Positive Integer ◽  
Dirichlet Series ◽  
Weyl Group ◽  
Algebraic Number ◽  
Type A ◽  
Algebraic Number Field ◽  
Basis Vector ◽  
Roots Of Unity ◽  
Finite Set ◽  
Multiple Dirichlet Series

This chapter describes Type A Weyl group multiple Dirichlet series. It begins by defining the basic shape of the class of Weyl group multiple Dirichlet series. To do so, the following parameters are introduced: Φ‎, a reduced root system; n, a positive integer; F, an algebraic number field containing the group μ‎₂ₙ of 2n-th roots of unity; S, a finite set of places of F containing all the archimedean places, all places ramified over a ℚ; and an r-tuple of nonzero S-integers. In the language of representation theory, the weight of the basis vector corresponding to the Gelfand-Tsetlin pattern can be read from differences of consecutive row sums in the pattern. The chapter considers in this case expressions of the weight of the pattern up to an affine linear transformation.

scholarly journals Leavitt path algebras satisfying a polynomial identity

2016 ◽  
Vol 15 (05) ◽  
pp. 1650084 ◽  
Author(s):  
Jason P. Bell ◽  
T. H. Lenagan ◽  
Kulumani M. Rangaswamy
Keyword(s):  
Polynomial Identity ◽  
Finite Graph ◽  
Infinite Graphs ◽  
Leavitt Path Algebra ◽  
Arbitrary Graph ◽  
Graph Theoretic ◽  
Path Algebras ◽  
Leavitt Path Algebras ◽  
Dimension One ◽  
Kirillov Dimension

Leavitt path algebras [Formula: see text] of an arbitrary graph [Formula: see text] over a field [Formula: see text] satisfying a polynomial identity are completely characterized both in graph-theoretic and algebraic terms. When [Formula: see text] is a finite graph, [Formula: see text] satisfying a polynomial identity is shown to be equivalent to the Gelfand–Kirillov dimension of [Formula: see text] being at most one, though this is no longer true for infinite graphs. It is shown that, for an arbitrary graph [Formula: see text], the Leavitt path algebra [Formula: see text] has Gelfand–Kirillov dimension zero if and only if [Formula: see text] has no cycles. Likewise, [Formula: see text] has Gelfand–Kirillov dimension one if and only if [Formula: see text] contains at least one cycle, but no cycle in [Formula: see text] has an exit.

scholarly journals ON A VARIATION OF A CONGRUENCE OF SUBBARAO

2012 ◽  
Vol 93 (1-2) ◽  
pp. 85-90 ◽  
Author(s):  
ANDREJ DUJELLA ◽  
FLORIAN LUCA
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Euler Function ◽  
Prime Factors ◽  
Finite Set ◽  
Positive Integers ◽  
Sum Of Divisors

AbstractWe study positive integers $n$ such that $n\phi (n)\equiv 2\hspace{0.167em} {\rm mod}\hspace{0.167em} \sigma (n)$, where $\phi (n)$ and $\sigma (n)$ are the Euler function and the sum of divisors function of the positive integer $n$, respectively. We give a general ineffective result showing that there are only finitely many such $n$ whose prime factors belong to a fixed finite set. When this finite set consists only of the two primes $2$ and $3$ we use continued fractions to find all such positive integers $n$.

Generalized Regular Sequence and Finiteness of Local Cohomology Modules

2008 ◽  
Vol 15 (03) ◽  
pp. 457-462 ◽  
Author(s):  
A. Mafi ◽  
H. Saremi
Keyword(s):  
Positive Integer ◽  
Local Ring ◽  
Local Cohomology ◽  
Regular Sequence ◽  
Natural Homomorphism ◽  
Finitely Generated ◽  
Noetherian Local Ring ◽  
Local Cohomology Modules ◽  
Finite Set ◽  
Cohomology Modules

Let R be a commutative Noetherian local ring, 𝔞 an ideal of R, and M a finitely generated generalized f-module. Let t be a positive integer such that [Formula: see text] and t > dim M - dim M/𝔞M. In this paper, we prove that there exists an ideal 𝔟 ⊇ 𝔞 such that (1) dim M - dim M/𝔟M = t; and (2) the natural homomorphism [Formula: see text] is an isomorphism for all i > t and it is surjective for i = t. Also, we show that if [Formula: see text] is a finite set for all i < t, then there exists an ideal 𝔟 of R such that dim R/𝔟 ≤ 1 and [Formula: see text] for all i < t.

Affine algebras of Gelfand-Kirillov dimension one are PI

1985 ◽  
Vol 97 (3) ◽  
pp. 407-414 ◽  
Author(s):  
L. W. Small ◽  
J. T. Stafford ◽  
R. B. Warfield
Keyword(s):  
Polynomial Identity ◽  
Prime Radical ◽  
Finitely Generated ◽  
Finite Module ◽  
Affine Algebras ◽  
Dimension One ◽  
Kirillov Dimension ◽  
Algebra Over A Field

The aim of this paper is to prove:Theorem.Let R be an affine (finitely generated) algebra over a field k and of Gelfand-Kirillov dimension one. Then R satisfies a polynomial identity. Consequently, if N is the prime radical of R, then N is nilpotent and R/N is a finite module over its Noetherian centre.

Extrema of a Class of Functions on a Finite Set

1974 ◽  
Vol 26 (4) ◽  
pp. 806-819
Author(s):  
Kenneth W. Lebensold
Keyword(s):  
Positive Integer ◽  
Traveling Salesman Problem ◽  
Traveling Salesman ◽  
Polygonal Path ◽  
Finite Set ◽  
Vertex Set ◽  
The Traveling Salesman Problem ◽  
The Cost ◽  
Special Case ◽  
Class Of Functions

In this paper, we are concerned with the following problem: Let S be a finite set and Sm* ⊂ 2S a collection of subsets of S each of whose members has m elements (m a positive integer). Let f be a real-valued function on S and, for p ∊ Sm*, define f(P) as Σs∊pf (s). We seek the minimum (or maximum) of the function f on the set Sm*.The Traveling Salesman Problem is to find the cheapest polygonal path through a given set of vertices, given the cost of getting from any vertex to any other. It is easily seen that the Traveling Salesman Problem is a special case of this system, where S is the set of all edges joining pairs of points in the vertex set, Sm* is the set of polygons, each polygon has m elements (m = no. of points in the vertex set = no. of edges per polygon), and f is the cost function.

scholarly journals Equisum Partitions of Sets of Positive Integers

Algorithms ◽  
2019 ◽  
Vol 12 (8) ◽  
pp. 164
Author(s):  
Eggleton
Keyword(s):  
Positive Integer ◽  
Initial Interval ◽  
Finite Set ◽  
Positive Integers

Let V be a finite set of positive integers with sum equal to a multiple of the integer b. When does V have a partition into b parts so that all parts have equal sums? We develop algorithmic constructions which yield positive, albeit incomplete, answers for the following classes of set V, where n is a given positive integer: (1) an initial interval a∈Z+:a≤n; (2) an initial interval of primes p∈P:p≤n, where P is the set of primes; (3) a divisor set d∈Z+:d|n; (4) an aliquot set d∈Z+:d|n, d<n. Open general questions and conjectures are included for each of these classes.

Lattice Paths in E3 With Diagonal Steps

1967 ◽  
Vol 10 (5) ◽  
pp. 653-658 ◽  
Author(s):  
D.R. Stocks
Keyword(s):  
Positive Integer ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Lattice Paths ◽  
Planar Lattice ◽  
Finite Set ◽  
Three Dimensional Space

Moser and Zayachkowski [1] have discussed certain planar lattice paths from (0, 0) to (m, n). In this note we consider analogous paths in three dimensional space. For basic definitions see reference [2]. Throughout this note each of m, n and k is a positive integer and if S is a finite set, |S| will denote the number of elements of S.Each path under consideration here is such that each of its steps is of one of the following types:

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.

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