Properties of Standard n-Ideals of a Lattice

0219-175X

convex sublattice ; standard n-ideal ; neutral element ; homomorphism n-kernel ; boolean algebra

Springer Online Journal Archives 1860-2000

Mathematics

Abstract An n-ideal of a lattice L is a convex sublattice containing a fixed element n ∈ L and it is called standard if it is a standard element of the lattice of n-ideals In(L). In this paper we have shown that, for a neutral element n of a lattice L, the principal n-ideal 〈a〉n of a lattice L is a standard n-ideal if and only if a ∨ n is standard and a ∧ n is dual standard. We have also shown that if n is a neutral element and (n] and [n) are relatively complemented, then every homomorphism n-kernels of L is a standard n-ideal and every standard n-ideal is the n-kernel of precisely one congruence relation. Finally, we have shown that, for a relatively complemented lattice L with 0 and 1, C(L) is a Boolean algebra if and only if every standard n-ideal of L is a principal n-ideal.

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