Affine Parts of Algebraic Theories II

1978 ◽  
Vol 30 (02) ◽  
pp. 231-237
Author(s):  
J. R. Isbell ◽  
M. I. Klun ◽  
S. H. Schanuel
Keyword(s):  
Algebraic Theory ◽  
Finitely Generated ◽  
Relative Complexity ◽  
Binary Operations ◽  
Number Of Generators ◽  
Algebraic Theories ◽  
Minimum Number ◽  
Finitely Related ◽  
Affine Part

This paper concerns relative complexity of an algebraic theory T and its affine part A, primarily for theories TR of modules over a ring R. TR, AR and R itself are all, or none, finitely generated or finitely related. The minimum number of relations is the same for T R and AR. The minimum number of generators is a very crude invariant for these theories, being 1 for AR if it is finite, and 2 for TR if it is finite (and 1 ≠ 0 in R). The minimum arity of generators is barely less crude: 2 for TR} and 2 or 3 for AR (1 ≠ 0). AR is generated by binary operations if and only if R admits no homomorphism onto Z2.

scholarly journals Growth sequences of finitely generated groups II

1989 ◽  
Vol 40 (2) ◽  
pp. 323-329 ◽  
Author(s):  
A.G.R. Stewart ◽  
James Wiegold
Keyword(s):  
Finitely Generated ◽  
Direct Power ◽  
Finitely Generated Groups ◽  
Number Of Generators ◽  
Minimum Number

A study is made of the minimum number of generators of the n-th direct power of certain finitely generated groups.

A finitely generated, infinitely related group with trivial multiplicator

1971 ◽  
Vol 5 (1) ◽  
pp. 131-136 ◽  
Author(s):  
Gilbert Baumslag
Keyword(s):  
Metabelian Group ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Related Group ◽  
Finitely Related

We exhibit a 3-generator metabelian group which is not finitely related but has a trivial multiplicator.1. The purpose of this note is to establish the exitense of a finitely generated group which is not finitely related, but whose multiplecator is finitely generated. This settles negatively a question whichb has been open for a few years (it was first brought to my attention by Michel Kervaire and Joan Landman Dyer in 1964, but I believe it is somewhat older). The group is given in the follwing theorem.

scholarly journals Large algebraic theories with small algebras

1978 ◽  
Vol 19 (3) ◽  
pp. 371-380 ◽  
Author(s):  
Jan Reiterman
Keyword(s):  
Algebraic Theory ◽  
Proper Class ◽  
Algebraic Theories

The aim of the paper is to study the interrelation between several natural smallness conditions on an algebraic theory with a proper class of operations. The conditions concern the existence of sets of data determining algebras, homomorphisms, subalgebras, and congruences.

scholarly journals Commutator Length of Finitely Generated Linear Groups

2008 ◽  
Vol 2008 ◽  
pp. 1-5
Author(s):  
Mahboubeh Alizadeh Sanati
Keyword(s):  
Finite Group ◽  
Upper Bound ◽  
Quadratic Polynomial ◽  
Linear Groups ◽  
Number Of Generators ◽  
Derived Subgroup ◽  
Finite Linear Group ◽  
Minimum Number ◽  
Commutator Length ◽  
Finite Linear

The commutator length “” of a group is the least natural number such that every element of the derived subgroup of is a product of commutators. We give an upper bound for when is a -generator nilpotent-by-abelian-by-finite group. Then, we give an upper bound for the commutator length of a soluble-by-finite linear group over that depends only on and the degree of linearity. For such a group , we prove that is less than , where is the minimum number of generators of (upper) triangular subgroup of and is a quadratic polynomial in . Finally we show that if is a soluble-by-finite group of Prüffer rank then , where is a quadratic polynomial in .

The multi-region index of a knot

2020 ◽  
Vol 29 (04) ◽  
pp. 2050022
Author(s):  
Sarah Goodhill ◽  
Adam M. Lowrance ◽  
Valeria Munoz Gonzales ◽  
Jessica Rattray ◽  
Amelia Zeh
Keyword(s):  
Lower Bound ◽  
Crossing Number ◽  
Number Of Generators ◽  
Branched Cover ◽  
Minimum Number

Using region crossing changes, we define a new invariant called the multi-region index of a knot. We prove that the multi-region index of a knot is bounded from above by twice the crossing number of the knot. In addition, we show that the minimum number of generators of the first homology of the double branched cover of [Formula: see text] over the knot is strictly less than the multi-region index. Our proof of this lower bound uses Goeritz matrices.

scholarly journals Growth sequences of finite semigroups

1987 ◽  
Vol 43 (1) ◽  
pp. 16-20 ◽  
Author(s):  
James Wiegold ◽  
H. Lausch
Keyword(s):  
Real Number ◽  
Finite Semigroup ◽  
Piecewise Linear ◽  
The Other ◽  
Direct Power ◽  
Growth Sequence ◽  
Group Of Units ◽  
Number Of Generators ◽  
Finite Semigroups ◽  
Minimum Number

AbstractThe growth sequence of a finite semigroup S is the sequence {d(Sn)}, where Sn is the nth direct power of S and d stands for minimum generating number. When S has an identity, d(Sn) = d(Tn) + kn for all n, where T is the group of units and k is the minimum number of generators of S mod T. Thus d(Sn) is essentially known since d(Tn) is (see reference 4), and indeed d(Sn) is then eventually piecewise linear. On the other hand, if S has no identity, there exists a real number c > 1 such that d(Sn) ≥ cn for all n ≥ 2.

T-systems of certain finite simple groups

1993 ◽  
Vol 113 (1) ◽  
pp. 9-22 ◽  
Author(s):  
Martin J. Evans
Keyword(s):  
Free Group ◽  
Simple Groups ◽  
Finite Simple Groups ◽  
Finitely Generated ◽  
Finitely Generated Group ◽  
Number Of Generators

Let Fn be the free group of rank n freely generated by x1, x2,…, xn and write d(G) for the minimal number of generators of the finitely generated group G.

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