English: Depicts the relation xRy defined by x ≤ ⁠5/4⁠y on natural numbers.

R is quasitransitive: Let x ≤ ⁠5/4⁠y and y ≤ ⁠5/4⁠z, but y > ⁠5/4⁠x and z > ⁠5/4⁠y. Then x ≤ ⁠5/4⁠y < z ≤ ⁠5/4⁠z, but z > ⁠5/4⁠y ≥ y > ⁠5/4⁠x.

For this reason, R can be written as the disjoint union of some symmetric relation J and some transitive relation P, and P can be chosen minimal with that property. The minimal P can be obtained by defining xPy by x < ⁠4/5⁠y for natural numbers x, y.

In the picture, xRy holds if the entry in line x, column y is not a red "·". If this entry is a green "P", even xPy holds. If this entry is a blue "I", "T", or "=", even xJy holds.

Given x, the set of numbers "indistinguishable" from x, i.e. { y: xJy }, equals the set of natural numbers in the interval [⁠4/5⁠x...⁠5/4⁠x]; thus it grows arbitrarily large if x is chosen appropriately. In the picture, the blue parts of horizontal slices grow arbitrarily wide.

Nevertheless P is a semiorder: It is asymmetric (semiorder axiom 1), since x < ⁠4/5⁠y < ⁠16/25⁠x is impossible for natural numbers. It satisfies semiorder axiom 2, since w < ⁠4/5⁠x and x ≥ ⁠4/5⁠y and y ≥ ⁠4/5⁠x and y < ⁠4/5⁠z implies w < ⁠4/5⁠x ≤ y < ⁠4/5⁠z. It satisfies semiorder axiom 3, since for x < ⁠4/5⁠y and y < ⁠4/5⁠z and arbitrary w, we have either w < ⁠4/5⁠z, meaning wPz, or w ≥ ⁠4/5⁠z, implying x < ⁠4/5⁠y < ⁠16/25⁠z ≤ ⁠4/5⁠w, that is xPw.

The union of P and all pairs that are incomparable with respect to P yields R again.