Geodesic
Natural extension of a congruence of a lattice to its lattice of convex sublattices
Archivum mathematicum, Tome 47 (2011) no. 2, pp. 133-138 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 $CS(L)$.
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 $CS(L)$.
Classification : 06B10, 06B20
Keywords: lattice of convex sublattices of a lattice; congruence relation; representable congruence relation 
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     title = {Natural extension of a congruence of a lattice to its lattice of convex sublattices},
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     pages = {133--138},
     year = {2011},
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}
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Bhatta, S. Parameshwara; Ramananda, H. S. Natural extension of a congruence of a lattice to its lattice of convex sublattices. Archivum mathematicum, Tome 47 (2011) no. 2, pp. 133-138. http://geodesic.mathdoc.fr/item/ARM_2011_47_2_a6/

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