Editing Partially ordered set (section)

== Intervals ==
{{See also|Interval (mathematics)}}

A '''convex set''' in a poset ''P'' is a subset {{mvar|I}} of ''P'' with the property that, for any ''x'' and ''y'' in {{mvar|I}} and any ''z'' in ''P'', if ''x'' ≤ ''z'' ≤ ''y'', then ''z'' is also in {{mvar|I}}. This definition generalizes the definition of [[interval (mathematics)|interval]]s of [[real number]]s. When there is possible confusion with [[convex set]]s of [[geometry]], one uses '''order-convex''' instead of "convex".

A '''convex sublattice''' of a [[lattice (order theory)|lattice]] ''L'' is a sublattice of ''L'' that is also a convex set of ''L''. Every nonempty convex sublattice can be uniquely represented as the intersection of a [[filter (mathematics)|filter]] and an [[ideal (order theory)|ideal]] of ''L''.

An '''interval''' in a poset ''P'' is a subset that can be defined with interval notation:
* For ''a'' ≤ ''b'', the ''closed interval'' {{closed-closed|''a'', ''b''}} is the set of elements ''x'' satisfying {{nowrap|''a'' ≤ ''x'' ≤ ''b''}} (that is, {{nowrap|''a'' ≤ ''x''}} and {{nowrap|''x'' ≤ ''b''}}). It contains at least the elements ''a'' and ''b''.
* Using the corresponding strict relation "<", the ''open interval'' {{open-open|''a'', ''b''}} is the set of elements ''x'' satisfying {{nowrap|''a'' < ''x'' < ''b''}} (i.e. {{nowrap|''a'' < ''x''}} and {{nowrap|''x'' < ''b''}}). An open interval may be empty even if {{nowrap|''a'' < ''b''}}.  For example, the open interval {{open-open|0, 1}} on the integers is empty since there is no integer {{mvar|x}} such that {{math|0 < {{var|x}} < 1}}.
* The ''half-open intervals'' {{closed-open|''a'', ''b''}} and {{open-closed|''a'', ''b''}} are defined similarly.

Whenever {{nowrap|''a'' ≤ ''b''}} does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold; for example, in the poset of divisors of 120, ordered by divisibility (see Fig.&nbsp;7b), the set {{mset|1, 2, 4, 5, 8}} is convex, but not an interval.

An interval {{mvar|I}} is bounded if there exist elements <math>a, b \in P</math> such that {{math|{{var|I}} ⊆ {{closed-closed|''a'', ''b''}}}}. Every interval that can be represented in interval notation is obviously bounded, but the converse is not true. For example, let {{math|1=''P'' = {{open-open|0, 1}} ∪ {{open-open|1, 2}} ∪ {{open-open|2, 3}}}} as a subposet of the real numbers. The subset {{open-open|1, 2}} is a bounded interval, but it has no [[infimum]] or [[supremum]] in&nbsp;''P'', so it cannot be written in interval notation using elements of&nbsp;''P''.

A poset is called [[Locally finite poset|locally finite]] if every bounded interval is finite. For example, the integers are locally finite under their natural ordering. The lexicographical order on the cartesian product <math>\N \times \N</math> is not locally finite, since {{math|(1, 2) ≤ (1, 3) ≤ (1, 4) ≤ (1, 5) ≤ ... ≤ (2, 1)}}.
Using the interval notation, the property "''a'' is covered by ''b''" can be rephrased equivalently as <math>[a, b] = \{a, b\}.</math>

This concept of an interval in a partial order should not be confused with the particular class of partial orders known as the [[interval order]]s.