compact element

Compact/finite elements


Let PP be a poset such that every directed subset of PP has a join; that is, PP is a dcpo. A compact element, or finite element, of PP is a compact object in PP regarded as a thin category; that is, homs out of it commute with these directed joins.

In other words, cPc \in P is compact precisely if for every directed subset {d i}\{d_i\} of PP we have

(c id i) i(cd i). (c \leq \bigvee_i d_i ) \Leftrightarrow \exists_i (c \leq d_i) \,.

Of course, the \Leftarrow part of this is automatic, so the real condition is the \Rightarrow part. In more elementary terms:

  • If cDc \leq \bigvee D for DD a directed subset, then cdc \leq d for some dDd \in D.

In the case where PP has a top element 11, we say that PP is compact if 11 is a compact element.


  • Given a set XX, the finite elements of its power set are precisely the (Kuratowski)-finite subsets of XX. (This is the origin of the term ‘finite element’.)

  • Given a topological space (or locale) XX, the compact elements of its frame of open subspaces are precisely the compact open subspaces of XX. (This is the origin of the term ‘compact element’.)

  • If RR is a (not necessarily commutative) ring, the lattice Idl(R)Idl(R) of two-sided ideals of RR is compact. Indeed, the top element is the ideal generated by 11, the multiplicative identity, and 1 iI i1 \in \bigvee_i I_i implies 1I i1 \in I_i for some index ii.

Revised on March 2, 2018 09:03:46 by Todd Trimble (