Is a total order also a partial order?

Question

Regarding task 1 of the current exercise I'm not sure if for a total order I also need to mark partial order as satisfied or not? In the script (picture below) it says that the total order extends the partial order as it has an ADDITIONAL criterion. However I wouldn't be surprised if you can not mark total order and partial order as satisfied at the same time because a order is either total or partial.

Same question for lattice and complete lattice. Should we write [0, 0, 0, 1, 1] if it's a complete lattice or [0, 0, 0, 0, 1]?

Answer 1

The definitions are meant to be read as non-overwriting. So an order stays partial even if it is total as well, and a lattice stays lattice even if it is a complete lattice as well.

Hence, do write [x,y,z,1,1] for complete lattices and not [x,y,z,0,1], also write [1,x,1,y,z] for total orders and not [1,x,0,y,z].

Comments

It is indeed an example for the insufficiencies of human language. In the mathematical sense, the definitions are precise and a relation that is a total order is necessarily also a partial order. However, in human language we tend towards classifying things in the most specific way and never call a total order partial since we rather consider partial and total as being opposites. The definitions are however not meant like that and rather have the non-overwriting sense as explained above.