Meet is Idempotent
Jump to navigation
Jump to search
Theorem
Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.
Then $\wedge$ is idempotent.
Proof
Let $a \in S$ be arbitrary.
Then:
\(\ds a \wedge a\) | \(=\) | \(\ds \inf \set {a, a}\) | Definition of Meet (Order Theory) | |||||||||||
\(\ds \) | \(=\) | \(\ds \inf \set a\) | Definition of Set | |||||||||||
\(\ds \) | \(=\) | \(\ds a\) | Infimum of Singleton |
Hence the result.
$\blacksquare$
Also see
This article is complete as far as it goes, but it could do with expansion. In particular: Needs to be phrased using a "partial operation" to remove the condition that $\wedge$ is a total operation You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by adding this information. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Expand}} from the code.If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page. |
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(a)}$