Some relations on the lattice of varieties of completely regular semigroups

Petrich, Mario

Bollettino della Unione matematica italiana, Série 8, 5B (2002) no. 2, pp. 265-278

Résumé

On the lattice $\mathcal{L}(\mathcal{CR})$ of varieties of completely regular semigroups considered as algebras with the binary multiplication and unary inversion within maximal subgroups, we study the relations $K_{l}$, $K$, $K_{r}$, $T_{l}$, $T$, $T_{r}$, $C$ and $L$. Here $K$ is the kernel relation, $T$ is the trace relation, $T_{l}$ and $T_{r}$ are the left and the right trace relations, respectively, $K_{p}=K \cap T_{p}$ for $p\in\{l,r \}$, $C$ is the core relation and $L$ is the local relation. We give an alternative definition for each of these relations $P$ of the form $$\mathcal{U}\ P\ \mathcal{V} \Leftrightarrow \mathcal{U} \cap \tilde{P} = \mathcal{V} \cap \tilde{P} \qquad (\mathcal{U}, \ \mathcal{V} \in \mathcal{L}(\mathcal{CR})),$$ for some subclasses $\tilde{P}$ of $\mathcal{CR}$. We also characterize the intersections of these relations and some joins within the lattice of equivalence relations on $\mathcal{L}(\mathcal{CR})$.