Exercise 28, § 1

Let G be a lattice ordered group with positive elements P. Two elements x and y of P are said to be equivalent if any element coprime to one is also coprime to the other : the equivalence classes under this relation are called the threads of P ; let $\overline{x}$ denote the thread containing x. a) Let $\overline{a}$ and $\overline{b}$ be threads ; let x, $x_1$ be elements of n and y, y, be elements of $\overline{b}$; show that if every element coprime to x is also coprime to y, then every element coprime to $x_1$ is also coprime to $y_1$. Define a relation in this way between a and b, written $a \geq b$; show that this is an ordering of the set F of threads. b) Show that F is a lattice under this ordering, and that

(use Cor. 3 of VI, p. 15). Show that F has a smallest element, namely the thread $\overline{0} = \{0\}$. c) Call two threads a, b coprime if $\inf(\overline{a}, \overline{b}) = \overline{0}$. Show that if a and b are two threads $\neq \overline{0}$, and if $\overline{a} < \overline{b}$, then there exists a thread $\overline{c}$, coprime to a, such that $\overline{c} < \overline{b}$. d) Call a thread m $\neq \overline{0}$ irreducible if it is a minimal element of the set of threads $\neq \overline{0}$. Show that an irreducible thread is totally ordered (if a and b are two elements of $\overline{m}$, consider the threads containing a $\inf(a, b)$ and $b - \inf(a, b)$ and use b)). Show also that the union of $\overline{m}$, $-\overline{m}$ and $\{0\}$ is a totally ordered subgroup II($\overline{m}$) of G. e) Given a family $(m_i)$ of distinct irreducible threads of G, show that the subgroup H generated by the union of the $\overline{m_i}$ is isomorphic to the direct sum of the groups H($\overline{m_i}$) (reduce to the case of a finite family, and argue by induction).

f) Show that in a product (resp. direct sum) of nonzero totally ordered groups, the elements of irreducible threads are those in which all the coordinates are zero except for one (which is positive). Deduce that an ordered group can be a product (resp. direct sum) of nonzero totally ordered groups in at most one way (in other words the factors are uniquely determined).