Internal Group Direct Product/Non-Examples

Examples of Structures which are not Internal Group Direct Products

Pointwise Addition on Continuous Real Functions on Unit Interval

Let $J \subseteq \R$ denote the closed unit interval $\closedint 0 1$.

Let $\map {\mathscr C} J$ denote the set of all continuous real functions from $J$ to $\R$.

Let $G = \struct {\map {\mathscr C} J, +}$ be the group formed on $\map {\mathscr C} J$ by pointwise addition.

Let $\struct {H, +}$ and $\struct {K, +}$ be the algebraic substructures of $G$ such that:

$H := \set {f \in G: \forall x \in J: \map f x \ge 0}$

$K := \set {f \in G: \forall x \in J: \map f x \le 0}$

Then, while $G$, $H$ and $K$ fulfil the $3$ conditions of Conditions for Internal Group Direct Product, $G$ is not the internal group direct product of $H$ and $K$.