Editing Ideal (order theory) (section)

== Definitions==

A subset {{mvar|I}} of a partially ordered set <math>(P, \leq)</math> is an '''ideal''', if the following conditions hold:<ref>{{harvtxt|Taylor|1999}}, [{{Google books|plainurl=y|id=iSCqyNgzamcC|page=141}} p. 141]: "A directed lower subset of a poset ''X'' is called an ideal"</ref><ref>{{cite book |last1=Gierz |first1=G. |last2=Hofmann |first2=K. H. |last3=Keimel |first3=K. |last4=Lawson |first4=J. D. |last5=Mislove |first5=M. W. |last6=Scott |first6=D. S. |title=Continuous Lattices and Domains |volume=93 |series=Encyclopedia of Mathematics and its Applications |publisher=Cambridge University Press |year=2003 |isbn=0521803381 |page=[https://archive.org/details/continuouslattic0000unse/page/3 3] |url=https://archive.org/details/continuouslattic0000unse/page/3 }}</ref>

# {{mvar|I}} is  [[non-empty]],
# for every ''x'' in {{mvar|I}} and ''y'' in ''P'', {{math|''y'' ≤ ''x''}} implies that ''y'' is in {{mvar|I}} &nbsp;({{mvar|I}} is a [[lower set]]),
# for every ''x'', ''y'' in {{mvar|I}}, there is some element ''z'' in {{mvar|I}}, such that {{math|''x'' ≤ ''z''}} and {{math|''y'' ≤ ''z''}} &nbsp;({{mvar|I}} is a [[directed set]]).

While this is the most general way to define an ideal for arbitrary posets, it was originally defined for [[Lattice (order)|lattice]]s only. In this case, the following equivalent definition can be given:
a subset {{mvar|I}} of a lattice <math>(P, \leq)</math> is an ideal [[if and only if]] it is a lower set that is closed under finite [[join and meet|joins]] ([[suprema]]); that is, it is nonempty and for all ''x'', ''y'' in {{mvar|I}}, the element <math>x \vee y</math> of ''P'' is also in {{mvar|I}}.{{sfn|Burris|Sankappanavar|1981|loc=Def. 8.2}}

{{anchor|order ideal}}A weaker notion of '''order ideal''' is defined to be a subset of a poset {{mvar|P}} that satisfies the above conditions 1 and 2. In other words, an order ideal is simply a [[lower set]]. Similarly, an ideal can also be defined as a "directed lower set".

The [[Duality (order theory)|dual]] notion of an ideal, i.e., the concept obtained by reversing all ≤ and exchanging <math>\vee</math> with <math>\wedge,</math> is a [[Filter (mathematics)|filter]].

[[Frink ideal]]s, [[pseudoideal]]s and [[pseudoideal|Doyle pseudoideals]] are different generalizations of the notion of a lattice ideal.

An ideal or filter is said to be '''proper''' if it is not equal to the whole set ''P''.{{sfn|Burris|Sankappanavar|1981|loc=Def. 8.2}}

The smallest ideal that contains a given element ''p'' is a {{em|{{visible anchor|principal ideal}}}} and ''p'' is said to be a {{em|{{visible anchor|principal element}}}} of the ideal in this situation. The principal ideal <math>\downarrow p</math> for a principal ''p'' is thus given by {{math|↓ ''p'' {{=}} {{mset|''x'' &isin; ''P'' | ''x'' ≤ ''p''}}}}.