Definition 6

— An element x of an ordered group G is said to be irreducible if it is a minimal element in the set of strictly positive elements of G.

Let x be an irreducible element of the ordered group G; if y is a positive element of G then the element inf(x, y), if it exists, can only be equal to x or to 0. Thus in a lattice ordered group G, every positive y is either greater than or coprime to the irreducible element x; in particular, two distinct irreducible elements are coprime.

(DIV) An element p of A is called irreducible if the ideal (p) is an irreducible element of the ordered group $\mathscr{P}^*$; this says that p is neither zero nor invertible, and that any element of $A$ which divides p is associate either to p or to 1. If $\mathscr{P}^*$ is lattice ordered, then every $a \in A$ is either coprime to p or a multiple of p.