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

## S. Parameshwara Bhatta and H. S. Ramananda

Address:

Department of Mathematics, Mangalore University, Mangalagangothri, 574 199, Karnataka State, INDIA

E-mail:

s_p_bhatta@yahoo.co.in

ramanandahs@gmail.com

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 $CS(L)$.

AMSclassification: primary 06B20; secondary 06B10.

Keywords: lattice of convex sublattices of a lattice, congruence relation, representable congruence relation.