Abstract: Suppose X_n={1,2,…,n}, with the usual order of natural number sequence, MC_n was the monotonic compression singular transformation semigroup on X_n. For any 3≤r≤n-1, we considered the semigroup MC(n,r)={α∈MC_n: |Im α|≤r}, and obtained the complete classification of the maximal subsemigroups of the semigroup MC(n,r) on X_n.

Key words: maximal subsemigroup, transformation semigroup, monotonic compression