Editing Completeness (order theory) (section)

===Finite completeness===

Further simple completeness conditions arise from the consideration of all non-empty [[finite set]]s. An order in which all non-empty finite sets have both a supremum and an infimum is called a [[lattice (order)|lattice]]. It suffices to require that all suprema and infima of ''two'' elements exist to obtain all non-empty finite ones; a straightforward [[mathematical induction|induction]] argument shows that every finite non-empty supremum/infimum can be decomposed into a finite number of binary suprema/infima. Thus the central operations of lattices are binary suprema <math>\vee</math> and infima {{nobreak|<math>\wedge</math>.}} It is in this context that the terms meet for <math>\wedge</math> and join for <math>\vee</math> are most common.

A poset in which only non-empty finite suprema are known to exist is therefore called a [[semilattice|join-semilattice]]. The dual notion is [[semilattice|meet-semilattice]].