# Classical (or less classical) examples of posets with incomparable elements

As classic examples of ordering relations for wich there are incomparable elements, I could give :

• the divisibility relation amongst integers : 5 and 7 are incomparable as to divisibility

• set inclusion : {a, b, c} and {a, b, d} are incomparable as to inclusion ( though they are not disjoint).

I think these two examples are the most common.

What other examples ( still classic, but more original) could be given?

• Real valued functions on any set by pointwise order. Any two functions that satisfy opposite inequalities at different points are incomparable. See also Riesz spaces. Symmetric or Hermitian matrices by positive definiteness, if $A-B$ is indefinite they are incomparable. – Conifold May 2 '19 at 9:38

## 1 Answer

Well, I guess you're right. But if you consider lattices, a lattice is distributive if and only if it does not contain the diamond and the pentagon lattice as a sub-lattice. These two lattices are given by posets with incomparable elements. See https://en.wikipedia.org/wiki/Distributive_lattice