Classifying word problems of finitely generated algebras via computable reducibility

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 [Formula: see text]-degrees). On the negative side, building on previous work of Kasymov and Khoussainov, we individuate a collection of [Formula: see text]-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 [Formula: see text]-degrees which embeds rich structures and is large in several reasonable ways.