Editing Complete lattice (section)

== Formal definition ==
A ''complete lattice'' is a [[partially ordered set]] (''L'', ≤) such that every [[subset]] ''A'' of ''L'' has both a [[greatest lower bound]] (the [[infimum]], or ''meet'') and a [[least upper bound]] (the [[supremum]], or ''join'') in (''L'', ≤).

The ''meet'' is denoted by <math>\bigwedge A</math>, and the ''join'' by <math>\bigvee A</math>.

In the special case where ''A'' is the [[empty set]], the meet of ''A'' is the [[greatest element]] of ''L''. Likewise, the join of the empty set is the [[least element]] of ''L''. Then, complete lattices form a special class of [[bounded lattice]]s.

=== Complete sublattices ===
A sublattice ''M'' of a complete lattice ''L'' is called a ''complete sublattice'' of ''L'' if for every subset ''A'' of ''M'' the elements <math>\bigwedge A</math> and <math>\bigvee A</math>, as defined in ''L'', are actually in ''M''.<ref>Burris, Stanley N., and H.P. Sankappanavar, H. P., 1981. ''[http://www.thoralf.uwaterloo.ca/htdocs/ualg.html A Course in Universal Algebra.]''  Springer-Verlag. {{isbn|3-540-90578-2}} (A monograph available free online).</ref>

If the above requirement is lessened to require only non-empty meet and joins to be in ''M'', the sublattice ''M'' is called a ''closed sublattice'' of ''L''.

=== Complete semilattices ===
The terms ''complete [[meet-semilattice]]'' or ''complete [[join-semilattice]]'' is another way to refer to complete lattices since arbitrary meets can be expressed in terms of arbitrary joins and vice versa (for details, see [[completeness (order theory)|completeness]]). 

Another usage of "complete meet-semilattice" refers to a meet-semilattice that is [[bounded complete]] and a [[complete partial order]]. This concept is arguably the "most complete" notion of a meet-semilattice that is not yet a lattice (in fact, only the top element may be missing).

See [[semilattice]]s for further discussion between both definitions. 

===Conditionally Complete Lattices===
A lattice is said to be "''conditionally complete''" if it satisfies [[Logical disjunction|either or both]] of the following properties:<ref>{{Cite web |last=Baker |first=Kirby |date=2010 |title=Complete Lattices |url=https://www.math.ucla.edu/~baker/222a/handouts/s_complete.pdf |access-date=8 June 2022 |website=UCLA Department of Mathematics}}</ref>

* Any subset bounded above has the [[least upper bound]].
* Any subset bounded below has the [[greatest lower bound]].