Which of the following is the correct definition of an anti chain ..
1). A subset S of a partially ordered set P is called an anti chain if no two elements of S are comparable ..
2). A subset S of a partially ordered set P is called an anti chain if no two DISTINCT elements of S are comparable. ..
If we take def 1 then in case of non strict partial order no non empty subset can be an anti chain .. because of reflexivity .. will that be right .. ?
Kindly help
Yes, you are correct. However, in some contexts it is easier to define partial orders as irreflexive and transitive (i.e. strict partial orders), in which case the two definitions indeed coincide.
You get the reverse situation with the definitions of chains, where "any two elements are comparable" is a definition which works for the reflexive case, but not for the strict case where you need to require them to be distinct.
(I will add that many times the 'distinct' part is implicitly assumed to get a cleaner definition regardless of context.)