Normal, simple and neutral congruences on lattices
Iqbalunnisa · Illinois Journal of Mathematics · 1966
IQBALUNNISAIt is known that the lattice of congruences of any lattice is a complete, Z-distributive lattice and hence is pseudo-complemented.Thus given any congruence 0 of a lattice L we can talk of its pseudo-complement (i.e., the pseudo-complement of 0 in 0(L) ).In this paper we characterize the pseudo- complement of any congruence on a lattice L and establish sets of necessary and sufficient conditions for a congruence 0 on L to be (i) normal and (ii) simple, in Theorems 1-3.This in turn enables us to give a characterization of weakly modular lattices in terms of its congruences (Theorem 4).Next we deal with neutral congruences and show in Theorem 5 that any congruence on a weakly complemented and dually weakly complemented lattice satisfying either chain condition is a neutral congruence.Further we show that a permutable congruence 0 on a lattice L with 0, 1 is complemented if and only if 0 is a principal neutral congruence on L (Theorem 6).Another theorem of this paper is" If L is a lattice with zero satisfying the ascending chain condition then there is a (1-1) correspondence between neutral ideals and congruences on L if and only if L is a direct union of simple lattices (Theorem 7).Making use of the characterization of weakly modular lattices of Theorem 4, we arrive at a necessary condition for a (1-1) correspond- ence between neutral ideals and congruences for general lattices in Theorem 8.We start with the definitions of some of the not too well known concepts used in the course of this paper.An element n of a lattice L is said to be a neutral element of L if it satisfies the following equalities"An element n of a lattice L satisfying conditions (ii) and (iii) is called a standard element of L.Any ideal I of L which is neutral (standard) considered as an element of I(L), the lattice of ideals of L is called a neutral (standard) ideal of L.An element a' of a lattice with zero is called the pseudocomplement of an element a in L if it satisfies the conditions" (i) aa' 0 and (ii) ax 0 ira- plies x _ a" for all x in L.An element of a lattice L is said to be normal if a" a, and is called simple if a-+a' 1.