Meet Absorbs Join
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Theorem
Let $\struct {S, \vee, \preceq}$ be a join semilattice.
Let $\wedge$ denote meet.
Then $\wedge$ absorbs $\vee$.
That is, for all $a, b \in S$:
- $a \wedge \paren {a \vee b} = a$
Proof
By Dual Pairs (Order Theory), we observe that the theorem statement is dual to that of Join Absorbs Meet.
The result follows by the Global Duality Principle.
$\blacksquare$
Duality
The dual to this theorem is Join Absorbs Meet.
Sources
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text {III}$: The Natural Numbers: $\S 14$: Orderings: Exercise $14.23 \ \text {(a)}$