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Let ( L, ≤ ) be a lattice, x, y, z ∈ L

I am unable to understand why ( x ∧ z ) ∧ ( y ∧ z ) = z ∧ ( y ∧ x ) is mentioned as fact in Y.N. Singh's "Mathematical Foundation of Computer Science" - Pg 157

Could I prove this somehow, or is it an axiom?

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Both sides equal the infimum of the set $\{x,y,z\}$. – Robin Chapman Nov 22 '10 at 21:21

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There are at least two ways to do this:

  • As Robin Chapman suggests, show that each side of the equation is just the infimum of the set $\{x,y,z\}$

  • Prove that $\wedge$ is an associative and commutative operation, and then use this to reduce the left side to $x \wedge y \wedge (z \wedge z)$ which is $x \wedge y \wedge z$ because $z \wedge z = z$.

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Well, I get it that $( x ∧ z ) ∧ ( y ∧ z ) = ( ∧ { x, z } ) ∧ ( ∧ { y, z } ) = ∧ { x, y, z }$ From this, I am comfortable saying that : $∧ { x, y, z } = ( x ∧ y ) ∧ ( x ∧ z ) ∧ ( y ∧ z )$ But from here I cannot proceed to show that this is equivalnet to $z ∧ ( y ∧ x )$ Even it were given, say, $y ≤ z, ( z ∧ y ) = y$ We could say, $∧ { x, y, z } = ( x ∧ y ) ∧ ( x ∧ z ) ∧ y$ But from here I cannot proceed to show that this is equivalnet to $z ∧ ( y ∧ x )$ – user3740 Nov 23 '10 at 0:14
I was able to show that both sides equal the infimum of the set $\{x,y,z\}$, but I do want to know how I can proceed from the information in my above comment. – user3740 Nov 23 '10 at 1:03

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