on the maximal subsemigroups and rank properties of certain semigroups of partial injective contractions of a finite chain

Denote [n] to be a finite n chain {1, 2, . . . , n}. let cin be the semigroup of partial injective contractions on [n]. de- note oddpn, odcin and ocin to be the semigroups of order- preserving order-decreasing partial isometries, order-preserving order- decreasing and order-preserving partial injective contractions, re- spectively. in this paper, we characterize all the maximal sub- semigroups of oddpn, odcin and ocin, respectively. we also characterize the regular elements, green’s relations, their starred analogues and rank properties of odcin.