Order affine completeness of lattices with Boolean congruence lattices

Kaarli, Kalle, and Kuchmei, Vladimir. "Order affine completeness of lattices with Boolean congruence lattices." Czechoslovak Mathematical Journal 57.4 (2007): 1049-1065. <http://eudml.org/doc/31182>.

@article{Kaarli2007,
abstract = {This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices $\{\mathbf \{L\}\}$ easily reduces to the case when $\{\mathbf \{L\}\}$ is a subdirect product of two simple lattices $\{\mathbf \{L\}\}_1$ and $\{\mathbf \{L\}\}_2$. Our main result claims that such a lattice is locally order affine complete iff $\{\mathbf \{L\}\}_1$ and $\{\mathbf \{L\}\}_2$ are tolerance trivial and one of the following three cases occurs: 1) $\{\mathbf \{L\}\}=\{\mathbf \{L\}\}_1\times \{\mathbf \{L\}\}_2$, 2) $\{\mathbf \{L\}\}$ is a maximal sublattice of the direct product, 3) $\{\mathbf \{L\}\}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.},
author = {Kaarli, Kalle, Kuchmei, Vladimir},
journal = {Czechoslovak Mathematical Journal},
keywords = {order affine completeness; congruences of lattices; tolerances of lattices; order affine completeness; congruences of lattices; tolerances of lattices},
language = {eng},
number = {4},
pages = {1049-1065},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Order affine completeness of lattices with Boolean congruence lattices},
url = {http://eudml.org/doc/31182},
volume = {57},
year = {2007},
}

TY - JOUR
AU - Kaarli, Kalle
AU - Kuchmei, Vladimir
TI - Order affine completeness of lattices with Boolean congruence lattices
JO - Czechoslovak Mathematical Journal
PY - 2007
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 57
IS - 4
SP - 1049
EP - 1065
AB - This paper grew out from attempts to determine which modular lattices of finite height are locally order affine complete. A surprising discovery was that one can go quite far without assuming the modularity itself. The only thing which matters is that the congruence lattice is finite Boolean. The local order affine completeness problem of such lattices ${\mathbf {L}}$ easily reduces to the case when ${\mathbf {L}}$ is a subdirect product of two simple lattices ${\mathbf {L}}_1$ and ${\mathbf {L}}_2$. Our main result claims that such a lattice is locally order affine complete iff ${\mathbf {L}}_1$ and ${\mathbf {L}}_2$ are tolerance trivial and one of the following three cases occurs: 1) ${\mathbf {L}}={\mathbf {L}}_1\times {\mathbf {L}}_2$, 2) ${\mathbf {L}}$ is a maximal sublattice of the direct product, 3) ${\mathbf {L}}$ is the intersection of two maximal sublattices, one containing $\langle 0,1\rangle $ and the other $\langle 1,0\rangle $.
LA - eng
KW - order affine completeness; congruences of lattices; tolerances of lattices; order affine completeness; congruences of lattices; tolerances of lattices
UR - http://eudml.org/doc/31182
ER -