Chain based lattices

Abstract

In recent years several weakenings of Post algebras have been studied. Among these have been P 0 "lattices by T. Traezyk, Stone lattice of order n by T. Katrinak and A. Mitschke, and P-algebras by the present authors. Each of these system is an abstraction from certain aspects of Post algebras, and no two of them are comparable. In the present paper, the theory of P 0 -lattices will be developed further and two new systems, called Pi-lattices and P 2 -lattices are introduced. These systems are referred to as chain based lattices. P 2 -lattices form the intersection of all three weakenings mentioned above. While P-algebras and weaker systems such as L-algebras, Heyting algebras, and P-algebras, do not require any distinguished chain of elements other than 0, 1, chain based lattices require such a chain.