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

S. Parameshwara Bhatta and H. S. Ramananda

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


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.