A Note on the Join of Varieties of Monoids with LI

Author Nathan Grosshans

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Nathan Grosshans
  • Fachbereich Elektrotechnik/Informatik, University of Kassel, Germany


I want to thank Thomas Place, who suggested the link between essentially-V stamps and V∨LI, but also Luc Segoufin who started the discussion with Thomas Place and encouraged me to write the present article. My thanks go as well to the anonymous referees for their helpful comments and suggestions. Finally, I want to mention that the introductions of Jean-Éric Pin’s future book on algebraic automata theory and of Marc Zeitoun’s works cited in the references have been helpful inspirations for my own introduction.

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Nathan Grosshans. A Note on the Join of Varieties of Monoids with LI. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 51:1-51:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In this note, we give a characterisation in terms of identities of the join of V with the variety of finite locally trivial semigroups LI for several well-known varieties of finite monoids V by using classical algebraic-automata-theoretic techniques. To achieve this, we use the new notion of essentially-V stamps defined by Grosshans, McKenzie and Segoufin and show that it actually coincides with the join of V and LI precisely when some natural condition on the variety of languages corresponding to V is verified. This work is a kind of rediscovery of the work of J. C. Costa around 20 years ago from a rather different angle, since Costa’s work relies on the use of advanced developments in profinite topology, whereas what is presented here essentially uses an algebraic, language-based approach.

Subject Classification

ACM Subject Classification
  • Theory of computation → Formal languages and automata theory
  • Theory of computation → Algebraic language theory
  • Varieties of monoids
  • join
  • LI


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