In a lecture of real analysis (this course is about Lebesgue measure) the lecture said:

For a set $X$ - $P(X)$ (the power set) is a compele lattice: For every $S\subseteq P(X)$ there exist $\cup S=\cup_{A\in S}A$ and $\cap > S=\cap_{A\in S}A$

I read in Wikipedia that a lattice is called complete if it have a supremum and an infimum, I don't know exactly what a lattice is, but I do know that $(P(X),\subset)$ is partially ordered set. However, It doesn't seem that $\cap S\leq S$ and that $S\leq\cup S$.

Can someone please explain (in simple words) the meaning of a complete lattice, why does $\cup S,\cap S$ are called the supremum and the infimum ? As far as I see it is not true that $\cap S\leq S$ and that $S\leq\cup S$ if we treat $(P(X),\subset)$ as a partially ordered set.

[note: I am sorry for the many typos, I am working on the PC farm, and the Lyx here doesn't have a spell checker]


$\leq$ isn't even defined on $P(X)$, eh? When the abstract lattice theory says $\leq$, the powerset lattice says $\subseteq$: in short, replace all instances of $\leq,\vee,\wedge$ with $\subseteq,\cup,\cap$ to translate from arbitrary lattices to lattices of sets.

You see then that for every $A\in S\subseteq P(X)$, $\cap S\subseteq A$ and $A\subseteq \cup S$. In an arbitrary lattice, instead of talking about $\cup$ and $\cap$ we talk about "join" and "meet," symbolized by $\vee$ and $\wedge$. Every lattice is a partially ordered set also required to have finite joins and meets, that is to say finite least upper and greatest lower bounds with respect to the lattice's partial order; the complete lattices are just those which have joins and meets of their infinite subsets as well.

So the powerset lattice is complete since every family of subsets of a set has an intersection and a union, in lattice-theoretic terminology a meet and a join, in other words a supremum and an infimum.

  • $\begingroup$ I don't understand...if $X=\{a.b.c\}$ and $S=\{\{a,b\},\{b,c\}\}$ then $\cap S=\{b\}$ and so it is not true that $\cap S\subset S$ $\endgroup$ – Belgi Oct 30 '12 at 8:09
  • $\begingroup$ That's just bad notation. What he means is, $\bigcap S$ is smaller than anything in $S$, and $\bigcup S$ is bigger than anything in $S$. $\endgroup$ – Zhen Lin Oct 30 '12 at 8:12
  • $\begingroup$ @ZhenLin - what is the meaning of the words smaller and bigger here ? $\endgroup$ – Belgi Oct 30 '12 at 8:13
  • $\begingroup$ Thanks for the comments-my original notation was terribly unclear. "smaller" and "bigger" here just mean $\subseteq$ and $\supseteq$. The point is, as I've edited in, that $\cap S$ is contained in every element of $S$ and similarly for $\cup S$: the lattice operations are applied to every element of a collection. $\endgroup$ – Kevin Carlson Oct 30 '12 at 8:23

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