File:Quasitransitive Even 5.gif

English: Depicts the relation xRy defined by (x < y and x+y is even) or (-5 < x-y < +5) on natural numbers.

R is quasitransitive, but doesn't satisfy any of axioms 1-3 of a semiorder.

Due to the former, 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 5 < x < y and x+y is even. The corresponding J can be obtained by defining xJy by -5 < x-y < +5.

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

A counter-example for semiorder axiom 2 is:

• 10 R 16,
• 16 and 23 are incomparable w.r.t. R,
• 23 R 29, but
• not 10 R 29.

A counter-example for semiorder axiom 3 is:

• 10 R 16,
• 16 R 28, but
• 23 is incomarable w.r.t. R to 10, to 16, and to 28.

Both examples still apply if R is replaced by P.

A counter-example for semiorder axiom 1 (asymmetry) is:

• 0 R 1, but also
• 1 R 0.

Since e.g. 16 is incomparable to 23, the relation R is not semi-connex, let alone connex; the same applies to P.