On the axiomatic rank of varieties generated by a semigroup or monoid with one defining relation

1989 ◽  
Vol 39 (1) ◽  
pp. 17-38 ◽  
Author(s):  
L. M. Shneerson
Keyword(s):  
Axiomatic Rank ◽  
Defining Relation

A non-cyclic one-relator group all of whose finite quotients are cyclic

1969 ◽  
Vol 10 (3-4) ◽  
pp. 497-498 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Normal Subgroup ◽  
Residually Finite ◽  
Defining Relation ◽  
Finite Quotient ◽  
Finite Quotients ◽  
Image Position

Let G be a group on two generators a and b subject to the single defining relation a = [a, ab]: . As usual [x, y] = x−1y−1xy and xy = y−1xy if x and y are elements of a group. The object of this note is to show that every finite quotient of G is cyclic. This implies that every normal subgroup of G contains the derived group G′. But by Magnus' theory of groups with a single defining relation G′ ≠ 1 ([1], §4.4). So G is not residually finite. This underlines the fact that groups with a single defining relation need not be residually finite (cf. [2]).

scholarly journals A theta relation in genus 4

2001 ◽  
Vol 161 ◽  
pp. 69-83 ◽  
Author(s):  
Eberhard Freitag ◽  
Manabu Oura
Keyword(s):  
Linear Combination ◽  
Modular Group ◽  
Remarkable Property ◽  
Graded Ring ◽  
Defining Relation

The 2gtheta constants of second kind of genusggenerate a graded ring of dimensiong(g +1)/2. In the caseg ≥3 there must exist algebraic relations. In genusg =3 it is known that there is one defining relation. In this paper we give a relation in the caseg =4. It is of degree 24 and has the remarkable property that it is invariant under the full Siegel modular group and whose Φ-image is not zero. Our relation is obtained as a linear combination of code polynomials of the 9 self-dual doubly-even codes of length 24.

The Axiomatic Rank of Levi Classes

2018 ◽  
Vol 57 (5) ◽  
pp. 381-391 ◽  
Author(s):  
S. A. Shakhova
Keyword(s):  
Axiomatic Rank

scholarly journals Quasirationality and prounipotent crossed modules

2019 ◽  
Vol 28 (13) ◽  
pp. 1940012
Author(s):  
A. M. Mikhovich
Keyword(s):  
Crossed Module ◽  
Defining Relation ◽  
Crossed Modules

We study quasirational (QR) presentations of (pro-[Formula: see text])groups, which contain aspherical presentations and their subpresentations, and also still mysterious pro-[Formula: see text]-groups with a single defining relation. Using schematization of QR-presentations and embedding of the rationalized module of relations into a diagram related to a certain prounipotent crossed module, we derive cohomological properties of pro-[Formula: see text]-groups with a single defining relation.

scholarly journals On a problem in the theory of ordered groups

1972 ◽  
Vol 6 (3) ◽  
pp. 435-438 ◽  
Author(s):  
Colin D. Fox
Keyword(s):  
Cambridge Philos ◽  
Orderable Group ◽  
Counter Example ◽  
Ordered Groups ◽  
Defining Relation

The group G presented on two generators a, c with the single defining relation a−1c2a = c2a2c2 [proposed by B.H. Neumann in 1949 (unpublished), discussed by Gilbert Baumslag in Proc. Cambridge Philos. Soc. 55 (1959)] has been considered as a possible example of an orderable group which can not be embedded in a divisible orderable group, contrary to the conjecture that no such examples exist. It is known from Baumslag's discussion that G can not be embedded in any divisible orderable group. However, it is shown in this note that G is not orderable, and thus is not a counter-example to the conjecture.

scholarly journals THREE-PARAMETER (TWO-SIDED) DEFORMATION OF HEISENBERG ALGEBRA

2012 ◽  
Vol 27 (21) ◽  
pp. 1250114 ◽  
Author(s):  
A. M. GAVRILIK ◽  
I. I. KACHURIK
Keyword(s):  
Deformation Parameter ◽  
Particle Number ◽  
Heisenberg Algebra ◽  
Unusual Property ◽  
Oscillator Algebra ◽  
Number Operator ◽  
The Third ◽  
Defining Relation ◽  
Deformed Oscillator ◽  
Deformed Oscillator Algebra

A three-parametric two-sided deformation of Heisenberg algebra (HA), with p, q-deformed commutator in the L.H.S. of basic defining relation and certain deformation of its R.H.S., is introduced and studied. The third deformation parameter μ appears in an extra term in the R.H.S. as pre-factor of Hamiltonian. For this deformation of HA we find novel properties. Namely, we prove it is possible to realize this (p, q, μ)-deformed HA by means of some deformed oscillator algebra. Also, we find the unusual property that the deforming factor μ in the considered deformed HA inevitably depends explicitly on particle number operator N. Such a novel N-dependence is special for the two-sided deformation of HA treated jointly with its deformed oscillator realizations.

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