Lattices Whose Finitely Generated n-ideals Form a Generalized Stone Lattice

Abstract

In this paper the authors have studied a lattice L whose set of finitely generated nideals Fn(L) form a generalized Stone lattice. They have shown that Fn(L) is generalized Stone if and only if x x L n n < >* ∨< >** = , which is also equivalent to( )* * * n n n n < x > ∩< y > =< x > ∨ < y > for all x, y∈L . < x >n denotes the principal n-ideal generated by x and * n < x > is the pseudo complement of < x >n in the lattice of n-ideals of L. They have also shown that F (L) n is generalized Stone if and only if P ∨ Q = L for any two minimal prime n-ideals P and Q of L.