Annulets in Almost Distributive Lattices

Abstract

We introduce the concept of annulets in an Almost Distributive lattice(ADL) $R$ with $0$. We characterize both generalized stone ADL and normal ADL in terms of their annulets. We characterize $\star$-ADLs by means of their annulets. It is proved that the lattice ${\mathcal{A}}_0(R)$ of all annulets of a generalized stone ADL $R$ is a relatively complemented sublattice of the lattice $\mathcal{I}(R)$ of all ideals of $R$. Finally, it is proved that ${\mathcal{A}}_0(R)$ is relatively complemented iff $R$ is sectionally $\star$-ADL.