If $(L, \vee, \wedge)$ be a lattice. Then we know that we can define an order on L by requiring $a \preceq b$ iff $a \wedge b = a$..
How can I show that any two elements in L say $x$ and $y$ have a supremum ( and an infimum )?
I can show that $w = x \vee y$ acts as upper bound for $x$ and $y$ .. but lack arguments to show that $w$ is the least upper bound for $x$ and $y$.
Kindly help me out ..
The relation $a\preceq b$ can also be described as $a\vee b=b$.
Since $a\preceq a\vee b$ and $b\preceq a\vee b$ we can recognize in $a\vee b$ an upper bound of $a$ and $b$.
If $a\preceq c$ and $b\preceq c$ then $c\vee\left(a\vee b\right)=\left(c\vee a\right)\vee b=c\vee b=c$ showing that $a\vee b\preceq c$.
This proves that $a\vee b$ serves as least upper bound of $a$ and $b$.