Say we have the set {a, b, c, d}. We can order it R = {(a,b),(a,c),(a,d)(b,c),(b,d),(c,d),(a,a),(b,b),(c,c)(d,d)} .
- Is it possible to create a partial order like this where every element is maximal?
- Or for an ordering where a is maximal but not a greatest element?
- I know the properties of a partial ordering, can you call it a partial order if 1 or 2 of the properties don't apply? or do all have to apply at once? i.e is R = {(a,a),(b,b),(c,c),(d,d)} a partial order since it is reflexive?
I'm new to partial ordering and from what I understand, 2 ordered pairs cannot be compared if they don't have the same elements, so could I partially order it like R = {(a,a),(b,b),(c,c),(d,d)} and say every element is maximal?
A partial order for which every element is maximal is an antichain.
A multipoint antichain has no greatest element.
All partial orders must be transitive, asymetric and reflexive.
The R you gave in 3 is an antichain.