Semilattice Induces Ordering

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

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


 * $a \preceq b$ $a \circ b = b$

Then $\preceq$ is an ordering.

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

Reflexivity
Since $\circ$ is idempotent, for all $a \in S$:


 * $a \circ a = a$

hence $a \preceq a$.

Thus $\preceq$ is reflexive.

Antisymmetry
Suppose that $a \preceq b$ and $b \preceq a$.

Then from the first relation:


 * $a \circ b = b$

and from the second:


 * $b \circ a = a$

Since $\circ$ is commutative, it follows that $a = b$.

Hence $\preceq$ is antisymmetric.

Transitivity
Suppose that $a \preceq b$ and $b \preceq c$.

Then:

Hence, $a \preceq c$.

Thus $\preceq$ is transitive.

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