Classifying word problems of finitely generated algebras via computable reducibility

Valentino Delle Rose, L. Mauro, A. Sorbi
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Abstract

We contribute to a recent research program which aims at revisiting the study of the complexity of word problems, a major area of research in combinatorial algebra, through the lens of the theory of computably enumerable equivalence relations (ceers), which has considerably grown in recent times. To pursue our analysis, we rely on the most popular way of assessing the complexity of ceers, that is via computable reducibility on equivalence relations, and its corresponding degree structure (the c-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of c-degrees of ceers which cannot be realized by the word problem of any finitely generated algebra of finite type. On the positive side, we show that word problems of finitely generated semigroups realize a collection of c-degrees which embeds rich structures and is large in several reasonable ways.
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利用可计算约化性对有限生成代数的词问题进行分类
我们为最近的一个研究项目做出了贡献,该项目旨在通过可计算可数等价关系(ceers)理论的视角,重新审视词问题的复杂性研究,这是组合代数的一个主要研究领域,该理论在最近得到了相当大的发展。为了继续我们的分析,我们依赖于最流行的评估ceers复杂性的方法,即通过等价关系的可计算约约性及其相应的度结构(c度)。在消极方面,在Kasymov和Khoussainov之前的工作的基础上,我们对c度的集合进行了个体化,这些集合不能通过任何有限生成的有限类型代数的字问题来实现。在积极的方面,我们证明了有限生成半群的词问题实现了c度的集合,该集合以几种合理的方式嵌入了丰富的结构并且很大。
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