scholarly journals Abelian torsion groups having a minimal system of generators

1961 ◽  
Vol 98 (3) ◽  
pp. 527-527 ◽  
Author(s):  
Samir A. Khabbaz
Keyword(s):  
Minimal System ◽  
Torsion Groups ◽  
Minimal System Of Generators

Implosive graphs: Square-free monomials on symbolic Rees algebras

2016 ◽  
Vol 16 (08) ◽  
pp. 1750145 ◽  
Author(s):  
A. Flores-Méndez ◽  
I. Gitler ◽  
E. Reyes
Keyword(s):  
Structural Properties ◽  
Monomial Ideal ◽  
Minimal System ◽  
Rees Algebra ◽  
Rees Algebras ◽  
Vertex Set ◽  
Minimal System Of Generators

Let [Formula: see text] be the edge monomial ideal of a graph [Formula: see text], whose vertex set is [Formula: see text]. [Formula: see text] is implosive if the symbolic Rees algebra [Formula: see text] of [Formula: see text] has a minimal system of generators [Formula: see text] where [Formula: see text] are square-free monomials. We give some structural properties of implosive graphs and we prove that they are closed under clique-sums and odd subdivisions. Furthermore, we prove that universally signable graphs are implosive. We show that odd holes, odd antiholes and some Truemper configurations (prisms, thetas and even wheels) are implosive. Moreover, we study excluded families of subgraphs for the class of implosive graphs. In particular, we characterize which Truemper configurations and extensions of odd holes and antiholes are minimal nonimplosive.

scholarly journals Schubert presentation of the cohomology ring of flag manifolds

2015 ◽  
Vol 18 (1) ◽  
pp. 489-506 ◽  
Author(s):  
Haibao Duan ◽  
Xuezhi Zhao
Keyword(s):  
Lie Group ◽  
Maximal Torus ◽  
Cohomology Ring ◽  
Minimal System ◽  
Schubert Calculus ◽  
Flag Manifolds ◽  
Generators And Relations ◽  
Integral Cohomology ◽  
Minimal System Of Generators ◽  
Connected Lie Group

Let $G$ be a compact connected Lie group with a maximal torus $T$. In the context of Schubert calculus we present the integral cohomology $H^{\ast }(G/T)$ by a minimal system of generators and relations.

Classification of Demushkin Groups

1967 ◽  
Vol 19 ◽  
pp. 106-132 ◽  
Author(s):  
John P. Labute
Keyword(s):  
Topological Group ◽  
Normal Subgroup ◽  
Bilinear Form ◽  
Minimal System ◽  
List Type ◽  
Type Number ◽  
Cup Product ◽  
Minimal System Of Generators ◽  
Closed Normal Subgroup

A pro-p-group G is said to be a Demushkin group if(1)dimFp H1(G, Z/pZ) < ∞,(2)dimFp H2(G, Z/pZ) = 1,(3)the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro-p-group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F9 (F, F); cf. §1.4.

On the BP-homology of 2e × 2e

2008 ◽  
Vol 3 (3) ◽  
pp. 409-435
Author(s):  
Leticia Zárate
Keyword(s):  
Formal Group ◽  
Minimal System ◽  
Easy Consequence ◽  
Annihilator Ideal ◽  
Formal Group Law ◽  
Minimal System Of Generators ◽  
Divisibility Properties ◽  
Group Law

AbstractWe study υ0- and υ1-divisibility properties of the [2e]-series associated to the universal 2-typical formal group law. This allows us to identify elements annihilating the toral class τ in BP*(2e × 2e). We conjecture that these elements form a minimal system of generators of the annihilator ideal of τ. This would provide a Landweber-type presentation for the BP-homology of 2e × 2e from which the relation hom:dimBP (Z2e × Z2e) = 2 would be an easy consequence.

scholarly journals Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

Mathematics ◽  
2021 ◽  
Vol 9 (24) ◽  
pp. 3204
Author(s):  
Manuel B. Branco ◽  
Isabel Colaço ◽  
Ignacio Ojeda
Keyword(s):  
Minimal System ◽  
Toric Ideal ◽  
Monomial Curves ◽  
Minimal System Of Generators ◽  
Positive Integers ◽  
The Ideal

Let a,b and n>1 be three positive integers such that a and ∑j=0n−1bj are relatively prime. In this paper, we prove that the toric ideal I associated to the submonoid of N generated by {∑j=0n−1bj}∪{∑j=0n−1bj+a∑j=0i−2bj∣i=2,…,n} is determinantal. Moreover, we prove that for n>3, the ideal I has a unique minimal system of generators if and only if a<b−1.

scholarly journals Minimal Systems of Binomial Generators for the Ideals of Certain Monomial Curves

2021 ◽  
Author(s):  
Manuel Baptista Branco ◽  
Isabel Colaço ◽  
Ignacio Ojeda
Keyword(s):  
Minimal System ◽  
Toric Ideal ◽  
Monomial Curves ◽  
Minimal System Of Generators ◽  
Positive Integers ◽  
The Ideal

Let $a, b$ and $n &gt; 1$ be three positive integers such that $a$ and $\sum_{j=0}^{n-1} b^j$ are relatively prime. In this paper, we prove that the toric ideal $I$ associated to the submonoid of $\mathbb{N}$ generated by $\{\sum_{j=0}^{n-1} b^j\} \cup \{\sum_{j=0}^{n-1} b^j + a\, \sum_{j=0}^{i-2} b^j \mid i = 2, \ldots, n\}$ is determinantal. Moreover, we prove that for $n &gt; 3$, the ideal $I$ has a unique minimal system of generators if and only if $a &lt; b-1$.

scholarly journals Homological Invariants of Local Rings

1963 ◽  
Vol 22 ◽  
pp. 219-227 ◽  
Author(s):  
Hiroshi Uehara
Keyword(s):  
Local Ring ◽  
Betti Numbers ◽  
Residue Field ◽  
Unit Element ◽  
Minimal System ◽  
Local Rings ◽  
Homology Algebra ◽  
Homological Invariants ◽  
Minimal System Of Generators ◽  
The Relationship

In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

$\mathscr A$-generators for the polynomial algebra of five variables in degree $5(2^t - 1) + 6.2^t$

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Positive Integer ◽  
Vector Space ◽  
Polynomial Algebra ◽  
Minimal System ◽  
Steenrod Algebra ◽  
Arbitrary Positive Integer ◽  
Left Module ◽  
Minimal System Of Generators ◽  
Set Up ◽  
Hit Problem

Let $P_s:= \mathbb{F}_2[x_1,x_2,\ldots ,x_s] = \bigoplus_{n\geqslant 0}(P_s)_n$ be the polynomial algebra viewedas a graded left module over the mod 2 Steenrod algebra, $\mathscr A.$ The grading is by the degree of the homogeneous terms $(P_s)_n$ of degree $n$ in the variables $x_1, x_2, \ldots, x_s$ of grading $1.$ We are interested in the {\it hit problem}, set up by F.P. Peterson, of finding a minimal system of generators for $\mathscr A$-module $P_s.$ Equivalently, we want to find a basis for the $\mathbb F_2$-graded vector space $\mathbb F_2\otimes_{\mathscr A} P_s.$ In this paper, we study the hit problem in the case $s=5$ and the degree $n = 5(2^t-1) + 6.2^t$ with $t$ an arbitrary positive integer.

scholarly journals Proportionally modular affine semigroups

2018 ◽  
Vol 17 (01) ◽  
pp. 1850017 ◽  
Author(s):  
J. I. García-García ◽  
M. A. Moreno-Frías ◽  
A. Vigneron-Tenorio
Keyword(s):  
Minimal System ◽  
Numerical Semigroups ◽  
Diophantine Inequalities ◽  
Generating Sets ◽  
Affine Semigroups ◽  
Affine Semigroup ◽  
Minimal System Of Generators

This work introduces a new kind of semigroup of [Formula: see text] called proportionally modular affine semigroup. These semigroups are defined by modular Diophantine inequalities and they are a generalization of proportionally modular numerical semigroups. We give an algorithm to compute their minimal generating sets, and we specialize when [Formula: see text]. For this case, we also provide a faster algorithm to compute their minimal system of generators, prove they are Cohen–Macaulay and Buchsbaum, and determinate their (minimal) Frobenius vectors. Besides, Gorenstein proportionally modular affine semigroups are characterized.

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