Join Absorbs Meet

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Theorem

Let $\struct {S, \wedge, \preceq}$ be a meet semilattice.

Let $\vee$ denote join.


Then $\vee$ absorbs $\wedge$.

That is, for all $a, b \in S$:

$a \vee \paren {a \wedge b} = a$


Proof

By Ordering in terms of Join, we have that:

$a \vee \left({a \wedge b}\right) = a$ if and only if $a \wedge b \preceq a$


The result thus follows from Meet Precedes Operands.

$\blacksquare$


Duality

The dual to this theorem is Meet Absorbs Join.