scholarly journals Generation of diagonal acts of some semigroups of transformations and relations

2005 ◽  
Vol 72 (1) ◽  
pp. 139-146 ◽  
Author(s):  
Peter Gallagher ◽  
Nik Ruškuc
Keyword(s):  
Symmetric Group ◽  
Binary Relations ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Infinite Set

The diagonal right (respectively, left) act of a semigroup S is the set S × S on which S acts via (x, y) s = (xs, ys) (respectively, s (x, y) = (sx, sy)); the same set with both actions is the diagonal bi-act. The diagonal right (respectively, left, bi-) act is said to be finitely generated if there is a finite set A ⊆ S × S such that S × S = AS1 (respectively, S × S = S1A, S × S = SlASl).In this paper we consider the question of finite generation for diagonal acts of certain infinite semigroups of transformations and relations. We show that the semi-groups of full transformations, partial transformations and binary relations on an infinite set each have cyclic diagonal right and left acts. The semigroup of full finite-to-one transformations on an infinite set has a cyclic diagonal right act but its diagonal left act is not finitely generated. The semigroup of partial injections on an infinite set has neither finitely generated diagonal right nor left act, but has a cyclic diagonal bi-act. The semigroup of bijections (symmetric group) on an infinite set does not have any finitely generated diagonal acts.

Finite presentability of generalized Bruck–Reilly ∗-extension of groups

2016 ◽  
Vol 09 (04) ◽  
pp. 1650090 ◽  
Author(s):  
Seda Oğuz ◽  
Eylem G. Karpuz
Keyword(s):  
Finite Subset ◽  
Identity Element ◽  
Finitely Generated ◽  
Finite Generation ◽  
Finite Set ◽  
Finite Presentability ◽  
Finitely Presented

In [Finite presentability of Bruck–Reilly extensions of groups, J. Algebra 242 (2001) 20–30], Araujo and Ruškuc studied finite generation and finite presentability of Bruck–Reilly extension of a group. In this paper, we aim to generalize some results given in that paper to generalized Bruck–Reilly ∗-extension of a group. In this way, we determine necessary and sufficent conditions for generalized Bruck–Reilly ∗-extension of a group, [Formula: see text], to be finitely generated and finitely presented. Let [Formula: see text] be a group, [Formula: see text] be morphisms and [Formula: see text] ([Formula: see text] and [Formula: see text] are the [Formula: see text]- and [Formula: see text]-classes, respectively, contains the identity element [Formula: see text] of [Formula: see text]). We prove that [Formula: see text] is finitely generated if and only if there exists a finite subset [Formula: see text] such that [Formula: see text] is generated by [Formula: see text]. We also prove that [Formula: see text] is finitely presented if and only if [Formula: see text] is presented by [Formula: see text], where [Formula: see text] is a finite set and [Formula: see text] [Formula: see text] for some finite set of relations [Formula: see text].

scholarly journals Finite representability of semigroups with demonic refinement

2021 ◽  
Vol 82 (2) ◽  
Author(s):  
Robin Hirsch ◽  
Jaš Šemrl
Keyword(s):  
Partial Order ◽  
Turing Machines ◽  
Binary Relations ◽  
Finite Representation ◽  
First Order ◽  
Finite Structures ◽  
Finite Base ◽  
Finite Set ◽  
Finite Representability ◽  
Representation Class

AbstractThe motivation for using demonic calculus for binary relations stems from the behaviour of demonic turing machines, when modelled relationally. Relational composition (; ) models sequential runs of two programs and demonic refinement ($$\sqsubseteq $$ ⊑ ) arises from the partial order given by modeling demonic choice ($$\sqcup $$ ⊔ ) of programs (see below for the formal relational definitions). We prove that the class $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) of abstract $$(\le , \circ )$$ ( ≤ , ∘ ) structures isomorphic to a set of binary relations ordered by demonic refinement with composition cannot be axiomatised by any finite set of first-order $$(\le , \circ )$$ ( ≤ , ∘ ) formulas. We provide a fairly simple, infinite, recursive axiomatisation that defines $$R(\sqsubseteq , ;)$$ R ( ⊑ , ; ) . We prove that a finite representable $$(\le , \circ )$$ ( ≤ , ∘ ) structure has a representation over a finite base. This appears to be the first example of a signature for binary relations with composition where the representation class is non-finitely axiomatisable, but where the finite representation property holds for finite structures.

scholarly journals Non-branching RCD(0,N) Geodesic Spaces with Small Linear Diameter Growth have Finitely Generated Fundamental Groups

2015 ◽  
Vol 58 (4) ◽  
pp. 787-798 ◽  
Author(s):  
Yu Kitabeppu ◽  
Sajjad Lakzian
Keyword(s):  
Diameter Growth ◽  
Fundamental Groups ◽  
Finitely Generated ◽  
Finite Generation ◽  
Alexandrov Spaces ◽  
Geodesic Spaces ◽  
Special Case ◽  
Linear Diameter

AbstractIn this paper, we generalize the finite generation result of Sormani to non-branching RCD(0, N) geodesic spaces (and in particular, Alexandrov spaces) with full supportmeasures. This is a special case of the Milnor’s Conjecture for complete non-compact RCD(0, N) spaces. One of the key tools we use is the Abresch–Gromoll type excess estimates for non-smooth spaces obtained by Gigli–Mosconi.

ON THE RANK OF ABELIAN VARIETIES OVER AMPLE FIELDS

2010 ◽  
Vol 06 (03) ◽  
pp. 579-586 ◽  
Author(s):  
ARNO FEHM ◽  
SEBASTIAN PETERSEN
Keyword(s):  
Finite Field ◽  
Abelian Variety ◽  
Galois Extension ◽  
Characteristic Zero ◽  
Rational Point ◽  
Abelian Varieties ◽  
Finitely Generated ◽  
Infinite Rank ◽  
Finite Set

A field K is called ample if every smooth K-curve that has a K-rational point has infinitely many of them. We prove two theorems to support the following conjecture, which is inspired by classical infinite rank results: Every non-zero Abelian variety A over an ample field K which is not algebraic over a finite field has infinite rank. First, the ℤ(p)-module A(K) ⊗ ℤ(p) is not finitely generated, where p is the characteristic of K. In particular, the conjecture holds for fields of characteristic zero. Second, if K is an infinite finitely generated field and S is a finite set of local primes of K, then every Abelian variety over K acquires infinite rank over certain subfields of the maximal totally S-adic Galois extension of K. This strengthens a recent infinite rank result of Geyer and Jarden.

scholarly journals Categorical Functional Data Analysis. The cfda R Package

Mathematics ◽  
2021 ◽  
Vol 9 (23) ◽  
pp. 3074
Author(s):  
Cristian Preda ◽  
Quentin Grimonprez ◽  
Vincent Vandewalle
Keyword(s):  
Functional Data ◽  
Multiple Correspondence Analysis ◽  
Real Data ◽  
Jump Process ◽  
R Package ◽  
Finite Basis ◽  
Data Set ◽  
Stochastic Jump ◽  
Finite Set ◽  
Infinite Set

Categorical functional data represented by paths of a stochastic jump process with continuous time and a finite set of states are considered. As an extension of the multiple correspondence analysis to an infinite set of variables, optimal encodings of states over time are approximated using an arbitrary finite basis of functions. This allows dimension reduction, optimal representation, and visualisation of data in lower dimensional spaces. The methodology is implemented in the cfda R package and is illustrated using a real data set in the clustering framework.

Curious Beginnings

2016 ◽  
Author(s):  
Joseph Mazur
Keyword(s):  
Historical Event ◽  
Mathematical Notation ◽  
Human Language ◽  
Mathematical Writing ◽  
Finite Set ◽  
And Algebra ◽  
Simple Equations ◽  
Infinite Set

This chapter traces the beginnings of mathematical notation. For tens of thousands of years, humans had been leaving signification marks in their surroundings, gouges on trees, footprints in hard mud, scratches in skin, and even pigments on rocks. A simple mark can represent a thought, indicate a plan, or record a historical event. Yet the most significant thing about human language and writing is that speakers and writers can produce a virtually infinite set of sounds, declarations, notions, and ideas from a finite set of marks and characters. The chapter discusses the emergence of the alphabet, counting, and mathematical writing. It also considers the discovery of traces of Sumerian number writing on clay tablets in caves from Europe to Asia, the use of Egyptian hieroglyphics, and algebra problems in the Rhind (or Ahmes) papyrus that presented simple equations without any symbols other than those used to indicate numbers.

Hopf Algebras from Branching Rules

1983 ◽  
Vol 35 (1) ◽  
pp. 177-192 ◽  
Author(s):  
P. Hoffman
Keyword(s):  
Abelian Group ◽  
Symmetric Group ◽  
Hopf Algebras ◽  
Homogeneous Element ◽  
Algebra Structure ◽  
Graded Algebra ◽  
Finitely Generated ◽  
Dimension Zero ◽  
Branching Rules

Below we work out the algebra structure of some Hopf algebras which arise concretely in restricting representations of the symmetric group to certain subgroups. The basic idea generalizes that used by Adams [1] for H*(BSU). The question arose in discussions with H. K. Farahat. I would like to thank him for his interest in the work and to acknowledge the usefulness of several stimulating conversations with him.1. Review and statement of results. A homogeneous element of a graded abelian group will have its gradation referred to as its dimension. In all such groups below there will be no non-zero elements with negative or odd dimension. A graded algebra (resp. coalgebra) will be associative (resp. coassociative), strictly commutative (resp. co-commutative) and in dimension zero will be isomorphic to the ground ring F, providing the unit (resp. counit). We shall deal amost entirely with F = Z or F = Z/p for a prime p; the cases F = 0 or a localization of Z will occur briefly. In every case, the component in each dimension will be a finitely generated free F-module, so dualization works simply.

scholarly journals Groups fixing graphas in switching classes

1982 ◽  
Vol 33 (1) ◽  
pp. 76-85
Author(s):  
Martin W. Liebeck
Keyword(s):  
Finite Group ◽  
Symmetric Group ◽  
Permutation Group ◽  
Abelian Groups ◽  
Finite Set ◽  
A Finite Group ◽  
Switching Class

AbstractA permutation group G on a finite set Ω is always exposable if whenever G stabilises a switching class of graphs on Ω, G fixes a graph in the switching class. Here we consider the problem: given a finite group G, which permutation representations of G are always exposable? We present solutions to the problem for (i) 2-generator abelian groups, (ii) all abelian groups in semiregular representations. (iii) generalised quaternion groups and (iv) some representations of the symmetric group Sn.

scholarly journals Asymptotic behaviour of ideals relative to injective modules over commutative Noetherian rings

1991 ◽  
Vol 34 (1) ◽  
pp. 155-160 ◽  
Author(s):  
H. Ansari Toroghy ◽  
R. Y. Sharp
Keyword(s):  
Asymptotic Behaviour ◽  
Noetherian Ring ◽  
Prime Ideals ◽  
Noetherian Rings ◽  
Finitely Generated ◽  
Commutative Noetherian Ring ◽  
Injective Modules ◽  
Finite Set ◽  
Attached Prime ◽  
Secondary Representation

LetEbe an injective module over the commutative Noetherian ringA, and letabe an ideal ofA. TheA-module (0:Eα) has a secondary representation, and the finite set AttA(0:Eα) of its attached prime ideals can be formed. One of the main results of this note is that the sequence of sets (AttA(0:Eαn))n∈Nis ultimately constant. This result is analogous to a theorem of M. Brodmann that, ifMis a finitely generatedA-module, then the sequence of sets (AssA(M/αnM))n∈Nis ultimately constant.

scholarly journals Generators and relations of Rees matrix semigroups

1999 ◽  
Vol 42 (3) ◽  
pp. 481-495 ◽  
Author(s):  
H. Ayik ◽  
N. Ruškuc
Keyword(s):  
Rees Matrix Semigroup ◽  
Finitely Generated ◽  
Finite Generation ◽  
Generators And Relations ◽  
Rees Matrix Semigroups ◽  
Finite Presentability ◽  
Matrix Semigroup ◽  
Finitely Presented ◽  
Matrix Semigroups ◽  
The Ideal

In this paper we consider finite generation and finite presentability of Rees matrix semigroups (with or without zero) over arbitrary semigroups. The main result states that a Rees matrix semigroup M[S; I, J; P] is finitely generated (respectively, finitely presented) if and only if S is finitely generated (respectively, finitely presented), and the sets I, J and S\U are finite, where U is the ideal of S generated by the entries of P.

Sign in / Sign up

Export Citation Format

Share Document