Editing Glossary of order theory (section)

== L ==

* '''[[Lattice (order)|Lattice]]'''. A lattice is a poset in which all non-empty finite joins (suprema) and meets (infima) exist.
* '''[[Least element]]'''. For a subset ''X'' of a poset ''P'', an element ''a'' of ''X'' is called the least element of ''X'', if ''a'' ≤ ''x'' for every element ''x'' in ''X''. The dual notion is called ''greatest element''.
* The '''length''' of a chain is the number of elements less one.  A chain with 1 element has length 0, one with 2 elements has length 1, etc.
* '''Linear'''. See ''total order''.
* '''[[Linear extension]]'''. A linear extension of a partial order is an extension that is a linear order, or total order.
* '''[[Complete Heyting algebra|Locale]]'''. A locale is a ''complete Heyting algebra''. Locales are also called ''frames'' and appear in [[Stone duality]] and [[pointless topology]].
* '''[[Locally finite poset]]'''. A partially ordered set ''P'' is ''locally finite'' if every interval [''a'', ''b''] = {''x'' in ''P'' | ''a'' ≤ ''x'' ≤ ''b''} is a finite set.
* '''[[Lower bound]]'''. A lower bound of a subset ''X'' of a poset ''P'' is an element ''b'' of ''P'', such that ''b'' ≤ ''x'', for all ''x'' in ''X''. The dual notion is called ''upper bound''.
* '''[[Lower set]]'''. A subset ''X'' of a poset ''P'' is called a lower set if, for all elements ''x'' in ''X'' and ''p'' in ''P'', ''p'' ≤ ''x'' implies that ''p'' is contained in ''X''. The dual notion is called ''upper set''.