Some relations on the lattice of varieties of completely regular semigroups

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 -