Natural extension of a congruence of a lattice to its lattice of convex sublattices

S. Parameshwara Bhatta; H. S. Ramananda

Archivum Mathematicum (2011)

  • Volume: 047, Issue: 2, page 133-138
  • ISSN: 0044-8753

Abstract

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Let L be a lattice. In this paper, corresponding to a given congruence relation Θ of L , a congruence relation Ψ Θ on C S ( L ) is defined and it is proved that 1. C S ( L / Θ ) is isomorphic to C S ( L ) / Ψ Θ ; 2. L / Θ and C S ( L ) / Ψ Θ are in the same equational class; 3. if Θ is representable in L , then so is Ψ Θ in C S ( L ) .

How to cite

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Bhatta, S. Parameshwara, and Ramananda, H. S.. "Natural extension of a congruence of a lattice to its lattice of convex sublattices." Archivum Mathematicum 047.2 (2011): 133-138. <http://eudml.org/doc/116541>.

@article{Bhatta2011,
abstract = {Let $L$ be a lattice. In this paper, corresponding to a given congruence relation $\Theta $ of $L$, a congruence relation $\Psi _\Theta $ on $CS(L)$ is defined and it is proved that 1. $CS(L/\Theta )$ is isomorphic to $CS(L)/\Psi _\Theta $; 2. $L/\Theta $ and $CS(L)/\Psi _\Theta $ are in the same equational class; 3. if $\Theta $ is representable in $L$, then so is $\Psi _\Theta $ in $CS(L)$.},
author = {Bhatta, S. Parameshwara, Ramananda, H. S.},
journal = {Archivum Mathematicum},
keywords = {lattice of convex sublattices of a lattice; congruence relation; representable congruence relation; lattice of convex sublattices; congruence relation; representable congruence relation},
language = {eng},
number = {2},
pages = {133-138},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Natural extension of a congruence of a lattice to its lattice of convex sublattices},
url = {http://eudml.org/doc/116541},
volume = {047},
year = {2011},
}

TY - JOUR
AU - Bhatta, S. Parameshwara
AU - Ramananda, H. S.
TI - Natural extension of a congruence of a lattice to its lattice of convex sublattices
JO - Archivum Mathematicum
PY - 2011
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 047
IS - 2
SP - 133
EP - 138
AB - Let $L$ be a lattice. In this paper, corresponding to a given congruence relation $\Theta $ of $L$, a congruence relation $\Psi _\Theta $ on $CS(L)$ is defined and it is proved that 1. $CS(L/\Theta )$ is isomorphic to $CS(L)/\Psi _\Theta $; 2. $L/\Theta $ and $CS(L)/\Psi _\Theta $ are in the same equational class; 3. if $\Theta $ is representable in $L$, then so is $\Psi _\Theta $ in $CS(L)$.
LA - eng
KW - lattice of convex sublattices of a lattice; congruence relation; representable congruence relation; lattice of convex sublattices; congruence relation; representable congruence relation
UR - http://eudml.org/doc/116541
ER -

References

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  1. Grätzer, G., General Lattice Theory, 2nd ed., Birkhäuser Verlag, 1998. (1998) MR1670580
  2. Grätzer, G., The Congruence of a Finite Lattice, A Proof by Picture Aproach, Birkhäuser Boston, 2006. (2006) MR2177459
  3. Lavanya, S., Parameshwara Bhatta, S., 10.1007/BF01190969, Algebra Universalis 35 (1996), 63–71. (1996) MR1360531DOI10.1007/BF01190969
  4. Parameshwara Bhatta, S., Ramananda, H. S., On ideals and congruence relations in trellises, Acta Math. Univ. Comenian. 2 (2010), 209–216. (2010) MR2745169

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