Editing Glossary of order theory (section)

== I ==

* '''[[Ideal (order theory)|Ideal]]'''. An '''[[Ideal (order theory)|ideal]]''' is a subset ''X'' of a poset ''P'' that is a directed lower set. The dual notion is called ''filter''.
* '''[[Incidence algebra]]'''. The '''[[incidence algebra]]''' of a poset is the [[associative algebra]] of all scalar-valued functions on intervals, with addition and scalar multiplication defined pointwise, and multiplication defined as a certain convolution; see [[incidence algebra]] for the details.
* '''[[Infimum]]'''. For a poset ''P'' and a subset ''X'' of ''P'', the greatest element in the set of lower bounds of ''X'' (if it exists, which it may not) is called the '''infimum''', '''meet''', or '''greatest lower bound''' of ''X''. It is denoted by inf ''X'' or <math>\bigwedge</math>''X''. The infimum of two elements may be written as inf{''x'',''y''} or ''x'' &and; ''y''. If the set ''X'' is finite, one speaks of a '''finite infimum'''.  The dual notion is called ''supremum''.
* '''[[Interval (mathematics)|Interval]]'''. For two elements ''a'', ''b'' of a partially ordered set ''P'', the ''interval'' [''a'',''b''] is the subset {''x'' in ''P'' | ''a'' ≤ ''x'' ≤ ''b''} of ''P''. If ''a'' ≤ ''b'' does not hold the interval will be empty.
*<span id="interval finite poset"></span>'''Interval finite poset'''. A partially ordered set ''P'' is '''interval finite''' if every interval of the form {x in P | x ≤ a} is a finite set.<ref>{{harvnb|Deng|2008|loc=p. 22}}</ref>
* '''Inverse'''.  See ''converse''.
* '''[[Irreflexive]]'''. A [[Relation (mathematics)|relation]] ''R'' on a set ''X'' is irreflexive, if there is no element ''x'' in ''X'' such that ''x R x''.
* '''Isotone'''.  See ''monotone''.