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.
@InProceedings{grosshans:LIPIcs.MFCS.2021.51, author = {Grosshans, Nathan}, title = {{A Note on the Join of Varieties of Monoids with LI}}, booktitle = {46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021)}, pages = {51:1--51:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-201-3}, ISSN = {1868-8969}, year = {2021}, volume = {202}, editor = {Bonchi, Filippo and Puglisi, Simon J.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2021.51}, URN = {urn:nbn:de:0030-drops-144918}, doi = {10.4230/LIPIcs.MFCS.2021.51}, annote = {Keywords: Varieties of monoids, join, LI} }
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