Semilattice Induces Ordering

Theorem
Let $\struct {S, \circ}$ be a semilattice.

Let $\RR$ be the relation on $S$ defined by, for all $a, b \in S$:


 * $a \mathrel \RR b$ $a \circ b = b$

Then $\RR$ is an ordering.

Proof
Let us verify that $\RR$ satisfies the three conditions for an ordering.

Reflexivity
From :
 * $\forall a \in S: a \circ a = a$

Hence by definition of $\RR$:


 * $\forall a \in S: a \mathrel \RR a = a$

So $\RR$ has been shown to be reflexive.

Antisymmetry
Let $a, b \in S$ such that $a \mathrel \RR b$ and $b \mathrel \RR a$.

So $\RR$ has been shown to be antisymmetric.

Transitivity
Let $a, b, c \in S$ such that $a \mathrel \RR b$ and $b \mathrel \RR c$.

Thus:

So $\RR$ has been shown to be transitive.

Having verified all three conditions, it follows that $\RR$ is an ordering.