Editing Glossary of order theory (section)

== M ==

* '''Maximal chain'''. A [[Total order#Chains|chain]] in a poset to which no element can be added without losing the property of being totally ordered. This is stronger than being a saturated chain, as it also excludes the existence of elements either less than all elements of the chain or greater than all its elements. A finite saturated chain is maximal if and only if it contains both a minimal and a maximal element of the poset.
* '''[[Maximal element]]'''. A maximal element of a subset ''X'' of a poset ''P'' is an element ''m'' of ''X'', such that ''m'' ≤ ''x'' implies ''m'' = ''x'', for all ''x'' in ''X''. The dual notion is called ''minimal element''.
* '''[[Greatest element|Maximum element]]'''. Synonym of greatest element. For a subset ''X'' of a poset ''P'', an element ''a'' of ''X'' is called the maximum element of ''X'' if ''x'' ≤ ''a'' for every element ''x'' in ''X''. A maxim''um'' element is necessarily maxim''al'', but the converse need not hold.
* '''Meet'''. See ''infimum''.
* '''[[Minimal element]]'''. A minimal element of a subset ''X'' of a poset ''P'' is an element ''m'' of ''X'', such that ''x'' ≤ ''m'' implies ''m'' = ''x'', for all ''x'' in ''X''. The dual notion is called ''maximal element''.
* '''[[Least element|Minimum element]]'''. Synonym of least element. For a subset ''X'' of a poset ''P'', an element ''a'' of ''X'' is called the minimum element of ''X'' if ''x'' ≥ ''a'' for every element ''x'' in ''X''. A minim''um'' element is necessarily minim''al'', but the converse need not hold.
* '''[[Monotone function|Monotone]]'''. A function ''f'' between posets ''P'' and ''Q'' is monotone if, for all elements ''x'', ''y'' of ''P'', ''x'' ≤ ''y'' (in ''P'') implies ''f''(''x'') ≤ ''f''(''y'') (in ''Q''). Other names for this property are ''isotone'' and ''order-preserving''. In [[Mathematical analysis|analysis]], in the presence of [[total order]]s, such functions are often called '''monotonically increasing''', but this is not a very convenient description when dealing with non-total orders. The dual notion is called ''antitone'' or ''order reversing''.