Partially ordered set: Difference between revisions

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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 sulatticesublattice''' of a [[lattice (order theory)|lattice]] ''L'' is a sublattice of a lattice''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 the lattice''L''.

An '''interval''' in a poset ''P'' is a subset that can be defined with interval notation (as follows). Every interval is a convex set, but the converse does not hold.:

* For ''a'' ≤ ''b'', the ''closed interval'' {{closed-closed|''a'', ''b''}} is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (that is, ''a'' ≤ ''x'' and ''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 ''a'' < ''x'' < ''b'' (i.e. ''a'' < ''x'' and ''x'' < ''b''). An open interval may be empty even if ''a'' < ''b''. For example, the open interval {{open-open|0, 1}} on the integers is empty since there areis no integersinteger {{mvar|Ix}} such that {{math|0 < {{var|Ix}} < 1}}.

* The ''half-open intervals'' {{closed-open|''a'', ''b''}} and {{open-closed|''a'', ''b''}} are defined similarly.

Whenever ''a'' ≤ ''b'' does not hold, all these intervals are empty. Every interval is a convex set, but the converse does not hold.

For ''a'' ≤ ''b'', the ''closed interval'' {{closed-closed|''a'', ''b''}} is the set of elements ''x'' satisfying ''a'' ≤ ''x'' ≤ ''b'' (that is, ''a'' ≤ ''x'' and ''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 ''a'' < ''x'' < ''b'' (i.e. ''a'' < ''x'' and ''x'' < ''b''). An open interval may be empty even if ''a'' < ''b''. For example, the open interval {{open-open|0, 1}} on the integers is empty since there are no integers {{mvar|I}} such that {{math|0 < {{var|I}} < 1}}.

The ''half-open intervals'' {{closed-open|''a'', ''b''}} and {{open-closed|''a'', ''b''}} are defined similarly.

Whenever ''a'' ≤ ''b'' does not hold, all these intervals are empty.

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 ''P'', so it cannot be written in interval notation using elements of ''P''.