  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.
