lattice-ordered group

Question

I am just starting to read the lattice-ordered group. I was trying to prove one of the elementary properties. Give an $\ell$-group $G$, for all $a, b$ and $c$ in $G$, $a + \textit{sup} (b, c) = \textit{sup} ((a + b),(a + c)) $. I have done the following.

$b \leq \textit{sup} (b, c)$ and $c \leq \textit{sup} (b, c)$. Then $ a + b \leq \textit{sup} (b, c) + a$ and $a + c \leq \textit{sup} (b, c) + a$. So, $ \textit{sup}((a + b), (a + c)) \leq a + \textit{sup} (b, c)$. How to show $a + \textit{sup} (b, c) \leq \textit{sup} ((a + b),(a + c))$ ?

Answer 1

To avoid any implication that the group is Abelian, I will write it multiplicatively. In that notation, and keeping track of the order of elements in a product, you have $b,c\le b\lor c$, so $ab\le a(b\lor c)$ and $ac\le a(b\lor c)$, and hence $ab\lor ac\le a(b\lor c)$.

HINT: One way to get the opposite inequality is to show that if $ab\lor ac\le x$, then $a(b\lor c)\le x$. If $ab\lor ac\le x$, then $ab\le x$ so $b\le a^{-1}x$. Similarly, ... ? (The conclusion is spoiler-protected; mouse-over to see it if you get stuck.)

Similarly, $c\le a^{-1}x$, and therefore $b\lor c\le a^{-1}x$. Now multiply on the left by $a$.