From small-overlap conditions to automatic semigroups

Automatic Rees matrix semigroups over categories

We consider the preservation of the properties of automaticity and prefix-automaticity in Rees matrix semigroups over semigroupoids and small categories. Some of our results are new or improve upon existing results in the single-object case of Rees matrix semigroups over semigroups.

Uniform decision problems in automatic semigroups

We consider various decision problems for automatic semigroups, which involve the provision of an automatic structure as part of the problem instance. With mild restrictions on the automatic structure, which seem to be necessary to make the problem well-defined, the uniform word problem for semigroups described by automatic structures is decidable. Under the same conditions, we show that one can also decide whether the semigroup is completely simple or completely zero-simple; in the case that it is, one can compute a Rees matrix representation for the semigroup, in the form of a Rees matrix to…

Small overlap monoids: the word problem

We develop a combinatorial approach to the study of semigroups and monoids with finite presentations satisfying small overlap conditions. In contrast to existing geometric methods, our approach facilitates a sequential left-right analysis of words which lends itself to the development of practical, efficient computational algorithms. In particular, we obtain a highly practical linear time solution to the word problem for monoids and semigroups with finite presentations satisfying the condition C(4), and a polynomial time solution to the uniform word problem for presentations satisfying the sam…

Small overlap monoids II: automatic structures and normal forms

We show that any finite monoid or semigroup presentation satisfying the small overlap condition C(4) has word problem which is a deterministic rational relation. It follows that the set of lexicographically minimal words forms a regular language of normal forms, and that these normal forms can be computed in linear time. We also deduce that C(4) monoids and semigroups are rational (in the sense of Sakarovitch), asynchronous automatic, and word hyperbolic (in the sense of Duncan and Gilman). From this it follows that C(4) monoids satisfy analogues of Kleene's theorem, and admit decision algorit…

幺半群与克隆上的自动连续性、唯一Polish拓扑及Zariski拓扑

Automatic continuity, unique Polish topologies, and Zariski topologies on monoids and clones

本文研究幺半群代数结构对相容拓扑的决定作用，证明多种著名幺半群具有自动连续性，并刻画其唯一Polish半群拓扑及Zariski拓扑。

小重叠幺半群中正规形式的显式算法

An explicit algorithm for normal forms in small overlap monoids

针对小重叠幺半群中现有正规形式线性时间算法系数过大的问题，提出一种时间复杂度为$O(|u|^2)$但系数更小的实用正规形式计算算法，同时将小重叠幺半群一致字问题的时间复杂度优化至$O(N\min\{|u|,|v|\})$。