Chain based lattices
George Epstein, Alfred Horn · Pacific Journal of Mathematics · 1974
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.Definitions are given in § 1.A P 0 -lattice is a bounded distributive lattice A which is generated by its center and a finite subchain containing 0 and 1.Such a subchain is called a chain base for A. The order of a P 0 -lattice A is the smallest number of elements in a chain base of A. In § 2, properties of P 0 -lattices are given which are used in later sections.If a P 0 -lattice A is a Heyting algebra, then it is shown in § 3, that there exists a unique chain base 0 = e 0 0. A P 0 -lattice with such a chain base is called a Pi-lattice.Every Pi-lattice of order n is a Stone lattice of order n.If a Pi-lattice is pseudo-supplemented then it is called a P 2 -lattice.It turns out that P 2 -lattices of order n are direct products of finitely many Post algebras whose maximum order is n.In § 4, properties of P 2 -lattices are studied.In § 5, equational axioms are given for P 2 -lattices.P 2 -lattices share many of the properties of Post algebras and have application to computer science.Among examples of P 2 -lattices are direct products of finitely many p-rings.These further remarks on P 2 -lattices are in § 6.In § 7, prime ideals in P 0 -lattices are studied.It is shown that the order of a P 0 -lattice is one more than the number of elements in a chain of prime ideals of maximum length.A characterization of P r lattices by properties of their prime ideals is given.Such a characterization of P 2 -lattices is also indicated.l DEFINITIONS.We use φ for the empty set.Let A be a distributive lattice which is bounded, that is, has a largest element 1 and a smallest element 0. The dual of A is denoted by A d .The DEFINITION 3.2.A P r lattice (A;e Of •--,e n _ 1 ) is a P 0 -lattice to-