Semiorder

Semiorder --- полупорядок.

The relation [math]\displaystyle{ P }[/math] is a semiorder if the following conditions hold:

(1) [math]\displaystyle{ P }[/math] is irreflexive;

(2) if [math]\displaystyle{ (x,y) \in P }[/math] and [math]\displaystyle{ (z,w) \in P }[/math], then [math]\displaystyle{ (x,w) \in P }[/math] or [math]\displaystyle{ (z,y) \in P }[/math];

(3) if [math]\displaystyle{ (x,y) \in P }[/math] and [math]\displaystyle{ (y,z) \in P }[/math], then [math]\displaystyle{ (x,w) \in P }[/math] or [math]\displaystyle{ (w,z) \in P }[/math].

semiorder is an interesting subclass of interval orders.