Natural Numbers under Min Operation forms Total Semilattice

Theorem
Let $\struct {\N, \wedge}$ denote the set of natural numbers $\N$ under the min operation:
 * $\forall a, b \in \N: a \wedge b := \min \set {a, b}$

Then $\struct {\N, \wedge}$ forms a total semilattice.

Proof
Taking the semilattice axioms in turn:

We have that:


 * $\min \set {a, b} = \begin {cases} a & : a \le b \\ b : b \le a \end {cases}$

Hence if $a, b \in \N$ then $a \wedge b \in \N$.

Thus $\struct {\N, \wedge}$ is closed.

Thus $\wedge$ is associative.

Thus $\wedge$ is commutative.

Thus $\wedge$ is idempotent.

All the semilattice axioms are thus seen to be fulfilled, and so $\struct {\N, \wedge}$ is a semilattice.

Totality
Let $T \subseteq S$.

Let $a, b\ in T$.

From Min Operation Equals an Operand:


 * $\forall a, b \in \N: \min \set {a, b} \in \set {a, b}$

Hence by definition of subset:


 * $\min \set {a, b} \in T$

Hence by definition $\struct {\N, \wedge}$ is a total semilattice.