论文标题
生长功能空间中的间隙和近似值
Gaps and approximations in the space of growth functions
论文作者
论文摘要
组合非交通性代数的一个重要问题是表征有限生成的代数的增长功能（等效地，半群或遗传语言）。 每个有限生成的无限维代数的增长函数在增加和近端。这些自然必要条件在多大程度上也足够的问题 - 特别是，至少足以实现足够快速的功能 - 由各种作者提出和研究，并吸引了一系列研究。 虽然每一个增加和潜在的功能都可以实现为生长函数，直到线性误差术语，但我们表明，任意迅速增加的潜入功能不等于任何代数的增长，从而解决了上述问题，从而解决了Zelmanov（Zelmanav）的问题（并由Alahmadi-Alsahmadi-Alsahmadi-alsulai-i-ainsulai-i-szelmanov提出了一个问题。这些可以解释为增长功能空间中的“孔”，从而积累了指数函数的顺序拓扑。 我们表明，存在其生长功能的单一代数和世袭语言编码了不可长期的单词的存在，而生长功能的代数编码了nilpotent理想的存在（在分级情况下）。在分级案例中，这是负面解决的另一个猜想。
An important problem in combinatorial noncommutative algebra is to characterize the growth functions of finitely generated algebras (equivalently, semigroups, or hereditary languages). The growth function of every finitely generated, infinite-dimensional algebra is increasing and submultiplicative. The question of to what extent these natural necessary conditions are also sufficient -- and in particular, whether they are sufficient at least for sufficiently rapid functions -- was posed and studied by various authors and has attracted a flurry of research. While every increasing and submultiplicative function is realizable as a growth function up to a linear error term, we show that there exist arbitrarily rapid increasing submultiplicative functions which are not equivalent to the growth of any algebra, thus resolving the aforementioned problem and settling a question posed by Zelmanov (and repeated by Alahmadi-Alsulami-Jain-Zelmanov). These can be interpreted as `holes' in the space of growth functions, accumulating to exponential functions in the order topology. We show that there exist monomial algebras and hereditary languages whose growth functions encode the existence of non-prolongable words, and algebras whose growth functions encode the existence of nilpotent ideals (in the graded case). This negatively solves another conjecture of Alahmadi-Alsulami-Jain-Zelmanov in the graded case.
