2017/03/15 by Lukas Fleischer, Fleischer, Lukas, Manfred Kufleitner +1
Computer Science · #Algorithms and Data Compression #FOS: Computer and information sciences #Formal Languages and Automata Theory (cs.FL) #Machine Learning and Algorithms #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1703.04941
openalex publication_date 2017/03/15 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the complexity of Green's relations when the semigroup is given by transformations on a finite set. Green's relations can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes then correspond to the strongly connected components. It is not difficult to show that, in the worst case, the number of equivalence classes is in the same order of magnitude as the number of elements. Another important parameter is the maximal length of a chain of components. Our main contribution is an exponential lower bound for this parameter. There is a simple construction for an arbitrary set of generators. However, the proof for constant alphabet is rather involved. Our results also apply to automata and their syntactic semigroups.