# A note on the join of varieties of monoids with LI

Abstract : In this note, we give a characterisation in terms of identities of the join of $\mathbf{V}$ with $\mathbf{LI}$ for several well-known varieties of monoids $\mathbf{V}$ by using classical algebraic-automata-theoretic techniques. To achieve this, we use the new notion of essentially-$\mathbf{V}$ stamps defined by Grosshans, McKenzie and Segoufin and show that it actually coincides with the join of $\mathbf{V}$ and $\mathbf{LI}$ precisely when some natural condition on the variety of languages corresponding to $\mathbf{V}$ is verified. This work is a kind of rediscovery of the work of J. C. Costa 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.
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Preprints, Working Papers, ...

https://hal.archives-ouvertes.fr/hal-03183838
Contributor : Nathan Grosshans <>
Submitted on : Monday, March 29, 2021 - 3:09:59 AM
Last modification on : Monday, March 29, 2021 - 4:35:21 PM

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### Identifiers

• HAL Id : hal-03183838, version 1
• ARXIV : 2103.15659

### Citation

Nathan Grosshans. A note on the join of varieties of monoids with LI. 2021. ⟨hal-03183838⟩

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