Some relations on the lattice of varieties of completely regular semigroups

Mario Petrich

Bollettino dell'Unione Matematica Italiana (2002)

  • Volume: 5-B, Issue: 2, page 265-278
  • ISSN: 0392-4041

Abstract

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On the lattice L C R 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 T p for p 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 U P V U P ~ = V P ~ ( U , V L ( C R ) ) , for some subclasses P ~ of C R . We also characterize the intersections of these relations and some joins within the lattice of equivalence relations on L C R .

How to cite

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Petrich, Mario. "Some relations on the lattice of varieties of completely regular semigroups." Bollettino dell'Unione Matematica Italiana 5-B.2 (2002): 265-278. <http://eudml.org/doc/195328>.

@article{Petrich2002,
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 $\mathcal\{L\}(\mathcal\{CR\})$.},
author = {Petrich, Mario},
journal = {Bollettino dell'Unione Matematica Italiana},
keywords = {complete congruences; lattices of varieties; varieties of completely regular semigroups},
language = {eng},
month = {6},
number = {2},
pages = {265-278},
publisher = {Unione Matematica Italiana},
title = {Some relations on the lattice of varieties of completely regular semigroups},
url = {http://eudml.org/doc/195328},
volume = {5-B},
year = {2002},
}

TY - JOUR
AU - Petrich, Mario
TI - Some relations on the lattice of varieties of completely regular semigroups
JO - Bollettino dell'Unione Matematica Italiana
DA - 2002/6//
PB - Unione Matematica Italiana
VL - 5-B
IS - 2
SP - 265
EP - 278
AB - 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 $\mathcal{L}(\mathcal{CR})$.
LA - eng
KW - complete congruences; lattices of varieties; varieties of completely regular semigroups
UR - http://eudml.org/doc/195328
ER -

References

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  7. PETRICH, M.- REILLY, N. R., Semigroups generated by certain operators on varieties of completely regular semigroups, Pacific J. Math., 132 (1988), 151-175.  Zbl0598.20061MR929587
  8. PETRICH, M.- REILLY, N. R., Operators related to E -disjunctive and fundamental completely regular semigroups, J. Algebra, 134 (1990), 1-27. Zbl0706.20043MR1068411
  9. PETRICH, M.- REILLY, N. R., Operators related to idempotent generated and monoid completely regular semigroups, J. Austral. Math. Soc., 49 (1990), 1-23.  Zbl0708.20019MR1054079
  10. REILLY, N. R., Varieties of completely regular semigroups, J. Austral. Math. Soc., 38 (1985), 372-393.  Zbl0572.20040MR779201
  11. REILLY, N. R.- ZHANG, S., Commutativity of operators on the lattice of existence varieties, Monatsh. Math., 123 (1997), 337-364. Zbl0870.20042MR1448576

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