On ideal theory for lattices

Historically ideal theory for lattices was developed by Hashimoto(10). He established that there is a one to one correspondence between ideals and congruence relations of a lattice L under which the ideal corresponding to a congruence relation is a whole congruence class under it if and only if L is a generalized Boolean algebra. His proof involved topological ideas which were later simplified, using lattice theoretic ideas, by Gratzer and Schmidt(8). Also Gratzer(6) introduced the notion of standard elements and ideals in lattices which were extensively studied by Gratzer and Schmidt(9). It was shown that standard ideals of lattices play a role somewhat similar to that of normal subgroups of groups or ideals of rings. Later, Fried and Schmidt(4) extended the notion of standard ideals of lattices to convex sublattices. Generalizations of some of these results to trellises (or also called weakly associative lattices) may be found in Bhatta and Ramananda(13) and Shashirekha(14).