Let (F)n be the full transformation semigroup on Xn=｛ 1,2,…,n ｝.Let 1 ≤ r≤ n,put(F)(n,r)={α∈(F)n∶iα=i,(˅)i∈｛1,2,…,r｝｝,it is obvious that the semigroup (F)(n,r) is subsemigroup of (F)n.In the paper,we study the core(L)(F)(n,r)=<E((F)(n,r))>of the semig-roup (F)(n,r),where E((F)(n,r))={α∈(F)(n,r)∶α2=α｝,by analyzing idempotents of the semigroup (F)(n,r),we prove that the rank and idempotent rank of semigroup(L)(F)(n,r)are both equal to(n-r)(n-r-1)/2+r(n-r)+1.
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