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## A Note on the Join of Varieties of Monoids with LI

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1
Nathan Grosshans

#### Abstract

In this note, we give a characterisation in terms of identities of the join of $\mathbf{V}$ with the variety of finite locally trivial semigroups $\mathbf{LI}$ for several well-known varieties of finite 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 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.

### Dates and versions

hal-03183838 , version 1 (29-03-2021)
hal-03183838 , version 2 (10-05-2021)
hal-03183838 , version 3 (27-08-2021)

### Identifiers

• HAL Id : hal-03183838 , version 3
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• DOI :

### Cite

Nathan Grosshans. A Note on the Join of Varieties of Monoids with LI. 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021), Aug 2021, Tallinn, Estonia. pp.51:1-51:16, ⟨10.4230/LIPIcs.MFCS.2021.51⟩. ⟨hal-03183838v3⟩

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