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A Note on the Join of Varieties of Monoids with LI Grosshans, Nathan Varieties of monoids join LI 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. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Nathan Grosshans 2021 Is Part Of LIPIcs, Volume 202, 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.MFCS.2021.51 urn:nbn:de:0030-drops-144918 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.51 eng https://creativecommons.org/licenses/by/4.0/legalcode