Meet Absorbs Join

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 $14.23 \ \text {(a)}$
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