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.

Abstract (VO)

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

Riassunto

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

Bibliographie

[1] K. Auinger, Complete congruences on lattices of (existence) varieties, Semigroups and Languages, Porto Conf. 1994, World Scientific (1996), 1-10. | MR | Zbl

[2] P. R. Jones, Mal'cev product of varieties of completely regular semigroups, J. Austral. Math. Soc., 42 (1987), 227-246. | MR | Zbl

[3] F. Pastijn, The lattice of completely regular varieties, J. Austral. Math. Soc., 49 (1990), 24-42. | MR | Zbl

[4] F. Pastijn-M. Petrich, Congruences on regular semigroups, Trans. Amer. Math. Soc., 295 (1986), 607-633. | MR | Zbl

[5] M. Petrich, Semigroups certain of whose subsemigroups have identities, Czechoslovak J. Math., 16 (1966), 186-198. | fulltext mini-dml | MR | Zbl

[6] M. Petrich, The Green relations approach to congruences on completely regular semigroups, Ann. Mat. Pura Appl. (iv), 167 (1994), 117-146. | MR | Zbl

[7] M. Petrich-N. R. Reilly, Semigroups generated by certain operators on varieties of completely regular semigroups, Pacific J. Math., 132 (1988), 151-175. | fulltext mini-dml | MR | Zbl

[8] M. Petrich-N. R. Reilly, Operators related to $E$-disjunctive and fundamental completely regular semigroups, J. Algebra, 134 (1990), 1-27. | MR | Zbl

[9] M. Petrich-N. R. Reilly, Operators related to idempotent generated and monoid completely regular semigroups, J. Austral. Math. Soc., 49 (1990), 1-23. | MR | Zbl

[10] N. R. Reilly, Varieties of completely regular semigroups, J. Austral. Math. Soc., 38 (1985), 372-393. | MR | Zbl

[11] N. R. Reilly-S. Zhang, Commutativity of operators on the lattice of existence varieties, Monatsh. Math., 123 (1997), 337-364. | MR | Zbl