I'm self studyhing from Peter Meyer-Nieberg's Banach Lattices, and I'm having some trouble with some of the very basic properties. So, what I have to work with at this point is the definition: We have a vector space E with a partial-order lattice (so all pairs have a sup and an inf)., such that addition of by a fixed vector and multiplication by nonnegative real scalars is preserved by the operation.
So, in his theorem 1.1.1 (i), he asserts that $\forall x,y,z\in E$, $$x+y=x\lor y+x\land y$$ $$x\lor y=-((-x)\land (-y))$$ $$x\lor y+z=(x+z)\lor(y+z)$$ $$x\land y+z=(x+z)\land (y+z)$$
The second one seems to be the usual inverse of a sup becomes the inf of the inverses, but does that automatically hold with an arbitrary space with just the Riesz properties listed above? For his proof, he defines $$w=(-y)\lor (-x)$$ $$v=x\lor y$$ and proceeds to show that $$w+x+y\ge v$$ I followed those steps, but I didn't see how that immediately gives the rest of the results claimed. I know later on he says that we will prove that all finite equality and inequality statements in a Riesz space holds if and only if the statement holds for real numbers, but we haven't gotten to that point yet.
You want to show that $$ x\wedge y = - ((-x)\vee (-y)) $$ So check that $$ x \leq x\wedge y \Rightarrow -(x\wedge y) \leq -x $$ and similarly with $y$ on the RHS. Hence $$ -(x\wedge y) \leq (-x)\vee (-y) $$ And for the reverse inequality, suppose $$ -x \geq z \text{ and } -y\geq z \Rightarrow -z \geq x \text{ and } -z \geq y $$ and hence $$ -z \geq x\wedge y \Rightarrow -(x\wedge y)\geq z $$ Hence, $-(x\wedge y)$ is the infimum of the set $\{-x,-y\}$.