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.