Yeah! THis forum rocks my socks
!!
Thank you in advance!
A chain is totally ordered subset. Because of the finite character every chain in a finite poset has a maximal term.
If that term is not maximal in the whole poset then the chain is not maximal.
What is the definition of each of: transitive closure, symmetric closure and reflexive closure of a relation?
I don't really have definitions... What I know is that if a relation R on a set S fails to have a certain property, you may be able to extend R to a relation R* on S that does have that property.
So for example for S = {0,1,2,3} and R={<0,1>,<0,2>,<1,1>,<1,3>,<2,2>,<3,0>} we know that R is not reflexive, symmetric or transitive. However, if we add the ordered pairs <0,0> and <3,3>, these pairs along with the original ones will give me the closure of R with respect to reflexivity.
Well I would assume that there must be some minimal additions to accomplish closure.
For example, just by uniting the diagonal with any relation makes it reflexive.
Also, I find this to be a trivial question for this reason.
Any relation that is reflexive, symmetric, and transitive is by definition an equivalence relation.