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Keywords:
lattice of convex sublattices of a lattice; congruence relation; representable congruence relation
lattice of convex sublattices of a lattice; congruence relation; representable congruence relation
Summary:
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 $CS(L)$.
References:
[1] Grätzer, G.: General Lattice Theory. 2nd ed., Birkhäuser Verlag, 1998. MR 1670580
[2] Grätzer, G.: The Congruence of a Finite Lattice, A Proof by Picture Aproach. Birkhäuser Boston, 2006. MR 2177459
[3] Lavanya, S., Parameshwara Bhatta, S.: A new approach to the lattice of convex sublattice of a lattice. Algebra Universalis 35 (1996), 63–71. DOI 10.1007/BF01190969 | MR 1360531
[4] Parameshwara Bhatta, S., Ramananda, H. S.: On ideals and congruence relations in trellises. Acta Math. Univ. Comenian. 2 (2010), 209–216. MR 2745169
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