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This work analyses the algebraic properties of the sub-semi group of partial Isometries (〖DP〗ₙ) and of order preserving partial isometries (〖ODP〗ₙ) of a finite chain Xₙ=1, 2, …n, with a symmetric inverse semigroup Iₙ defined on it. It also investigates the subsemigroups 〖DP〗ₙ and 〖ODP〗ₙ for cycle structure and shows that 〖ODP〗ₙ is a 0 -E- unitary inverse semigroup.
Udoaka et al. (Mon,) studied this question.