  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.
