Wednesday, September 21, 2011

Facts about distributivity

1Distributivity and complements¶

In a distributive lattice, all complements (if they exist) are unique. Proof: Suppose we have a distributive lattice consisting of $a, b, c \in L$, with $a \land b = \bot$, $a \lor b = \top$, $a\land c = \bot$, and $a \lor c = \top$. But $b = b \land \top = b \land (a \lor c) = (b \land a) \lor (b \land c) = b \land c$. Similarly, $c = b \land c$. Since both $b$ and $c$ are equal to the meet of $b$ and $c$, then we have that $b \leq c$ and $c \leq b$. So $b=c$ and so the complement to any element is unique.

Other topics covered during this class: hidden fives, failure of modularity, failure to stay awake¹