I have an order on $\mathbb R ^ \mathbb R: f \le g $ iff $\forall x \in \mathbb R: f(x) \le g(x)$. Now I have a function $\mathbb R ^ \mathbb R \to \mathbb R ^ \mathbb R: F (f)(x)= f(x)+f(-x)$. $\mathbb R ^ \mathbb R$ is a lattice, and I have to decide, if this function is lattice homomorphism or not.
As I understand it, $\inf\{f,g\} = h(x) = \inf\{f(x),g(x)\}$, so I have to deduce, if $F(h)(x)= h(x) + h(-x) = \inf \{f(x)+f(-x), g(x)+g(-x) \} = \inf \{F(f)(x), F(h)(x) \}$ or not. As $h(x) = \inf\{f(x),g(x)\}$, so is $h(-x) = \inf\{f(-x),g(-x)\}$, but after that, I am stuck. Is sum of infinums is infinum of sum? How is it for supremums? I am not sure how to continue.
Consider $f(x) = x$ and $g(x) = -x$. In this case $h(x) = \inf\{f(x), g(x)\} = -|x|.$
$$F(h)(x) = h(x) + h(-x) = -2|x|$$ $$\inf\{F(f)(x), F(g)(x)\} = \inf\{f(x) + f(-x), g(x) + g(-x)\} = \inf\{x - x, -x + x\} = 0.$$ So $F(\inf\{f, g\})(x) = -2|x| \neq 0 = \inf\{F(f)(x), F(g)(x)\}$ and hence $F$ is not a lattice homomorphism.