In recent years several
weakenings of Post algebras have been studied. Among these have been P0-lattices
by T. Traczyk, 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 P0-lattices will be developed further and two new
systems, called P1-lattices and P2-lattices are introduced. These systems are
referred to as chain based lattices. P2-lattices form the intersection of all
three weakenings mentioned above. While P-algebras and weaker systems
such as L-algebras, Heyting algebras, and B-algebras, do not require any
distinguished chain of elements other than 0,1, chain based lattices require such a
chain.
