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].