Annulets and α-ideals in a distributive lattice

Abstract

In a distributive lattice L with 0 the set of all ideals of the form (x]* can be made into a lattice A0(L) called the lattice of annulets of L. A 0(L) is a sublattice of the Boolean algebra of all annihilator ideals in L. While the lattice of annulets is no more than the dual of the so-called lattice of filets (carriers) as studied in the theory of l-groups and abstractly for distributive lattices in [1, section4] it is a useful notion in its own right. For example, from the basic theorem of [3] it follows that A 0(L) is a sublattice of the lattice of all ideals of L if and only if each prime ideal in L contains a unique minimal prime ideal.