Green’s Relations in Deterministic Finite Automata

  • Lukas FleischerEmail author
  • Manfred Kufleitner
Part of the following topical collections:
  1. Computer Science Symposium in Russia


Green’s relations are a fundamental tool in the structure theory of semigroups. They can be defined by reachability in the (right/left/two-sided) Cayley graph. The equivalence classes of Green’s relations then correspond to the strongly connected components. We study the complexity of Green’s relations in semigroups generated by transformations on a finite set. We 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 strongly connected 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 a constant size alphabet is rather involved. We also investigate the special cases of unary and binary alphabets. All these results are extended to deterministic finite automata and their syntactic semigroups.


Green’s relations Transformation semigroup Automaton Syntactic semigroup Complexity Bounds 



We thank the anonymous referees for valuable suggestions which helped to improve the presentation of the paper. We also thank Mikhail Volkov for inspiring comments which led to the addition of Section 5.


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Authors and Affiliations

  1. 1.FMIUniversity of StuttgartStuttgartGermany
  2. 2.Department of Computer ScienceLoughborough UniversityLoughboroughUK

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