On the semigroup of all injective orientation-preserving and order-decreasing partial transformations on a finite chain

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Let P O P D I n ( P O D I n ) be the semigroup consisting of all injective orientation-preserving (order-preserving) and order-decreasing transformations on the finite chain { 1 < ⋯ < n } . For 1 ≤ r ≤ n , let P O D I ( n , r ) = { α ∈ P O D I n : | im ( α ) | ≤ r } and P O P D I ( n , r ) = { α ∈ P O P D I n : | im ( α ) | ≤ r } . In this paper, we find the minimal generating sets, and so the ranks of P O D I ( n , r ) and P O P D I ( n , r ) . We also consider the nilpotent subsemigroup of P O D I ( n , r ) .