Growth of Generating Sets for Direct Powers of Classical Algebraic Structures

Abstract

Communicated by M. G. JacksonDedicated to the memory of Jim Wiegold. AbstractFor an algebraic structure A denote by d( A) the smallest size of a generating set for A, and let, where A n denotes a direct power of A. In this paper we investigate the asymptotic behaviour of the sequence d( A) when A is one of the classical structures-a group, ring, module, algebra or Lie algebra. We show that if A is finite then d( A) grows either linearly or logarithmically. In the infinite case constant growth becomes another possibility; in particular, if A is an infinite simple structure belonging to one of the above classes then d( A) is eventually constant. Where appropriate we frame our exposition within the general theory of congruence permutable varieties.2000 Mathematics subject classification: primary 08A40; secondary 16S15, 17B99, 20F05.

Growth rates of algebras, II: Wiegold dichotomy

“…James Wiegold proved in [7] that the growth rate of a finite perfect group is logarithmic (d A (n) ∈ Θ(log(n))), and that the growth rate of a finite imperfect group is linear (d A (n) ∈ Θ(n)). This result, herein called Wiegold dichotomy, was extended by Martyn Quick and Nik Ruškuc in [6] to several kinds of algebras that have underlying group structure. Namely, Quick and Ruškuc showed that a finite algebra A satisfies d A (n) ∈ Θ(log(n)) if A is a perfect ring, module, Lie algebra or k-algebra over a field k, and that d A (n) ∈ Θ(n) if A is an imperfect algebra of one of these types.…”

Growth Rates of Algebras, I: Pointed Cube Terms

“…What is really proved in [29] is that if Σ Grp is the set of identities axiomatizing the variety of groups and A is a finite algebra realizing Σ Grp , then A has a logarithmic growth rate if it is perfect and has a linear growth rate if it is imperfect. Although the results of [29] are stated for only a few specific varieties of group expansions, the results hold for any variety of group expansions.…”

“…Our work. We got interested in growth rates of finite algebras after reading [28,Remark 4.15], which states that At present no finite algebraic structure is known for which the d-sequence does not have one of logarithmic, linear or exponential growth. We found some of these missing algebras (Theorem 5.10).…”

On the growth of generating sets for direct powers of semigroups

“…In order to establish various examples of semigroups without identity, as required for (IG7)-(IG9), we utilise a construction introduced in [9] as a modification of an earlier construction by Byleen [2]. Let A and B be two (disjoint) countably infinite alphabets.…”

“…The results from above concerning groups largely carry over to other 'classical' algebraic structures, such as rings, associative and Lie algebras; this is the topic of [9]. The purpose of the present article is to present some results concerning the growth of d(S) for an infinite semigroup or monoid S. We mention in passing that recently d-sequences have (re)appeared in the context of Universal Algebra, in connection with quantified constraint satisfaction problems; see [5].…”