Local finiteness for Green relations in (I -)semigroup varieties

This work is devoted to the notion of local $\mathcal{K}$-finiteness for varieties, where $\mathcal{K}$ stands for any of the five Green's relations, with respect to varieties of semigroups and \emph{varieties of $I$-semigroups}, which are classes of algebras of type $(2,1)$ satisfying the identities $$x(yz)=(xy)z \, , \; xx'x=x \, , \; (x')' =x$$
(with $a\mapsto a'$ denoting the unary operation). Thus, for instance, both completely regular semigroup varieties and inverse semigroup varieties can be found within varieties of $I$-semigroups

For $\mathcal{K} \in \{\mathcal{H},\mathcal{L},\mathcal{R},\mathcal{D},\mathcal{J}\}$, we say that a semigroup is \emph{$\mathcal{K}$-finite} if it contains only finitely many (distinct) $\mathcal{K}$-classes and that a variety $\mathbf{V}$ of ($I$-)semigroups is \emph{locally $\mathcal{K}$-finite} if every finite generated semigroup from $\mathbf{V}$ is $\mathcal{K}$-finite.

In this talk, several properties of $\mathcal{K}$-finite semigroups will be described (namely, connections between the five types, conservation or loss under certain operators) and the lattices of varieties of semigroups and of varieties of $I$-semigroups studied and characterised with respect to these properties.