compact element

Let $X$ be a set and $\mathcal{T}$ be a topology on $X$ , a well-known concept is that of a compact set: a set $A$ is compact if every open cover of $A$ has a finite subcover. Another way of putting this, symbolically, is that if $$A\subseteq \bigcup \mathcal{S},$$ where $\mathcal{S}\subset \mathcal{T}$ , then there is a finite subset $\mathcal{F}$ of $\mathcal{S}$ , such that $$A\subseteq \bigcup \mathcal{F}.$$

A more general concept, derived from above, is that of a compact element in a lattice. Let $L$ be a lattice and $a\in L$ . Then $a$ is said to be compact if

whenever a subset $S$ of $L$ such that $\bigvee S$ exists and $a\le \bigvee S$ , then there is a finite subset $F\subset S$ such that $a\le \bigvee F$ .

If we let $\mathcal{D}$ to be the collection of closed subsets of $X$ , and partial order $\mathcal{D}$ by inclusion, then $\mathcal{D}$ becomes a lattice with meet and join defined by set theoretic intersection and union. It is easy to see that an element $A\in\mathcal{D}$ is a compact element iff $D$ is a compact closed subset in $X$ .

Here are some other common examples:

1. Let $C$ be a set and $2^C$ the subset lattice (power set) of $C$ . The compact elements of $2^C$ are the finite subsets of $C$ .
2. Let $V$ be a vector space and $L(V)$ be the subspace lattice of $V$ . Then the compact elements of $L(V)$ are exactly the finite dimensional subspaces of $V$ .
3. Let $G$ be a group and $L(G)$ the subgroup lattice of $G$ . Then the compact elements are the finitely generated subgroups of $G$ .
4. Note in all of the above examples, atoms are compact. However, this is not true in general. Let's construct one such example. Adjoin the symbol $\infty$ to the lattice $\mathbb{N}$ of natural numbers (with linear order), so that $n<\infty$ for all $n\in \mathbb{N}$ . So $\infty$ is the top element of $\mathbb{N}\cup\lbrace \infty\rbrace$ (and $1$ is the bottom element!). Next, adjoin a symbol $a$ to $\mathbb{N}\cup\lbrace \infty\rbrace$ , and define the meet and join properties with $a$ by
   • $a\vee n=\infty$ , $a\wedge n=1$ for all $n\in\mathbb{N}$ , and
   • $a\vee\infty =\infty$ , $a\wedge\infty = a$ .
   The resulting set $L=\mathbb{N}\cup\lbrace \infty,a\rbrace$ is a lattice where $a$ is a non-compact atom.

Remarks.

• As we have seen from the examples above, compactness is closely associated with the concept of finiteness, a compact element is sometimes called a finite element.
• Any finite join of compact elements is compact.
• An element $a$ in a lattice $L$ is compact iff for any directed subset $D$ of $L$ such that $\bigvee D$ exists and $a\le \bigvee D$ , then there is an element $d\in D$ such that $a\le d$ .
• As the last example indicates, not all atoms are compact. However, in an algebraic lattice, atoms are compact. The first three examples are all instances of algebraic lattices.
• A compact element may be defined in an arbitrary poset $P$ : $a\in P$ is compact iff $a$ is way below itself: $a\ll a$ .

Bibliography

1

G. Gierz, K. H. Hofmann, K. Keimel, J. D. Lawson, M. W. Mislove, D. S. Scott, Continuous Lattices and Domains, Cambridge University Press, Cambridge (2003).