Some relations on the lattice of varieties of completely regular semigroups

Petrich, Mario

Bollettino dell'Unione Matematica Italiana, Tome 5-A (2002), p. 265-278 / Harvested from Biblioteca Digitale Italiana di Matematica

Abstract

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 $\mathcal{L}(\mathcal{CR})$.

Résumé

Nel reticolo $\mathcal{L}(\mathcal{CR})$ delle varietà dei semigruppi completamente regolari, considerati come algebre con la moltiplicazione binaria e l'inversione unaria tra i sottogruppi massimali, si studiano le relazioni $K_{l}$, $K$, $K_{r}$, $T_{l}$, $T$, $T_{r}$, $C$ e $L$. Qui $K$ denota la relazione nucleo, $T$ la relazione traccia, $T_{l}$ e $T_{r}$ le relazioni traccia sinistra e destra rispettivamente, $K_{p} =K \cap T_{p}$ per $p\in\{l,r \}$, $C$ la relazione core ed $L$ la relazione locale. Viene data una definizione alternativa per ciascuna di queste relazioni $P$ nella forma $$\mathcal{U}\ P\ \mathcal{V} \Leftrightarrow \mathcal{U} \cap \tilde{P} = \mathcal{V} \cap \tilde{P} \qquad (\mathcal{U}, \ \mathcal{V} \in \mathcal{L}(\mathcal{CR})),$$ per alcune sottoclassi $\tilde{P}$ di $\mathcal{CR}$. Si caratterizzano inoltre le intersezioni di queste relazioni ed alcuni dei loro join nel reticolo delle equivalenze su $\mathcal{L}(\mathcal{CR})$.

Détail

Publié le : 2002-06-01

MR 1911191 | Zbl 1072.20067

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