Local finiteness for Green relations in (I )semigroup varieties
22/06/2016
International Conference of Semigroups and Automata Celebrating the 60th birthday of Jorge Almeida & Gracinda Gomes
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.
