Content deleted Content added
→Intervals: typo; use the name that was just introduced; mv theorem down after def; itemize def cases; better use x for numbers, I for intervals; singular is more clear |
→Intervals: made up a simple counter-example for "convex \not\implies interval" (if there is a convex sublattice that is not an interval, this should be given as example) |
||
Line 201:
* 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 example, in the poset of divisors of 120, ordered by divisibility (see Fig.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 ''P'', so it cannot be written in interval notation using elements of ''P''.
| |||