Non-isomorphic orders
Task number: 4533
Find infinitely many non-isomorphic orders on the set of positive integers.
Solution
Select any \(k\). Modify the standard ordering of positive integers so that the first \(k\) numbers are not comparable to any other numbers. Arrangements for different \(k\) will not be isomorphic because they have different numbers of minimal elements.
Variant
Find infinitely many non-isomorphic linear orders on the set of positive integers.Solution
If we want a linear ordering: we start from the standard ordering of rational numbers. Rational numbers are countable, we can transfer this ordering to positive integers numbers by bijection. Then we remove the numbers 0 to \(k\) from the order, arrange them linearly, and paste before the other numbers. This yields an order that has a minimum with \(k\) successors, the last of which no longer has a successor. Thus, these orders are non-isomorphic.
