nLab linear order

Disambiguation

A linear order could refer to either

• a total order

• a weakly total partial order

• a strict total order (i.e. DLO)

• a pseudo-order

Classically, the first two are the same, but constructively, the second notion is weaker than the first. Similarly, classically, the next two are the same, but constructively, the fourth notion is weaker than the third.

Classically, the first and third are complements of each other - and thus defining one is sufficient to defining the other, they still have different properties. In constructive mathematics, only the decidable such relations are complements of each other, and in general the two relations cannot be defined from the other.

category: disambiguation