Irreducible and Prime Elements1

In this paper L is a lattice and l is an element of L. The scheme NonUniqExD1 deals with a non empty relational structure A , a subset B of A , a non empty subset C of A , and a binary predicate P , and states that: There exists a function f from B into C such that for every element e of A if e ∈ B, then there exists an element u of A such that u ∈ C and u = f (e) and P [e,u] provided the parameters meet the following requirement: • For every element e of A such that e ∈ B there exists an element u of A such that u ∈ C and P [e,u]. Let L be a lattice, let A be a non empty subset of L, let f be a function from A into A, and let n be an element of N. Then f n is a function from A into A. Let L be a lattice, let C, D be non empty subsets of L, let f be a function from C into D, and let c be an element of C. Then f (c) is an element of L. Let L be a non empty poset. One can check that every chain of L is filtered and directed. One can verify that there exists a lattice which is strict, continuous, distributive, and lowerbounded. Next we state three propositions: