Total preorder

In Order Theory, a binary relation R is called total preorder on a set X if with the following properties:

• (Total)

Русский Русский x,and한 한 X:xRand andRx{displaystyle forall x,yin X:quad xRy;lor ;yRx}

• (Transitive)

Русский Русский x,and,z한 한 X:xRand∧ ∧ andRz⇒ ⇒ xRz{displaystyle forall x,y,zin X:quad xRy;land ;yRzquad Rightarrow quad xRz}

Example

In Social Choice Theory we can say the order of preferences is a total preorder. Suppose we have four candidates a, b, c and d. It can happen that:

• a be more preferred than b (a b){displaystyle (alesssim b)}
• a be more preferred than c (a c){displaystyle (alesssim c)}
• b more preferred than d (b d){displaystyle (blesssim d)}
• c more preferred than d (c d){displaystyle (clesssim d)}
• b is incomparable with c (b c c b){displaystyle (blesssim clor clesssim b)}

It's easy to verify {displaystyle lesssim } It's a total preorder.