The main purpose of this paper is to establish a relation between universality of
certain P-compactifications of a semitopological semigroup and their corresponding
enveloping semigroups. In particular, we show that if we take P to be the property that the enveloping semigroup of a compactification of a semitopological semigroup s is left simple, a group, or the trivial singleton semigroup, then the universal P- compactification of S would be the distal, the right simple, or the right zero
compactification , respectively