| Let Xn = {1,2, ? ? ?, n}(n≥3). The set IX consisting of all bijections between subsets of X (together with an empty mapping) is a semigroup, called the symmetric inverse semigroup, with the operation defined byDenote Sn the permutation group on Xn Let(n≥3). Then PDIn is a subsemigroup of In, called distance-preserving partial bijection semigroup on Xn. Let E be an equivalence on Xn. Then the setis a subsemigroup of In that preserve the equivalence E, and the setis also a subsemigroup of IE, called partial-preserving partial bijection semigroup on Xn.In this paper, we discuss some basic properties of the semigroups PDIn and PDIE2. The main results are given in the following.Theorem 2.2 PDIn is an inverse subsemigroup of In.Theorem 2.3 Letα,β∈PDIn. Then(1) (α,β)∈L (?) imα= imβ;(2) (α,β)∈R (?) domα= domβ;(3) (α,β)∈D (?)α≈β.Theorem 2.4 (?).Theorem 2.5 rank(PDIn) = n.Besides,we describe the minimal generating set and maximal inverse subsemigroup ofPDIn .In Chapter 3,we consider Green's relations of PDIE2, where E2 is a bi-equivalence on Xn(n≥5), namely, E2 = (A×A)∪(B×B)∪ΔX, where A and B axe disjoint subsets of Xn with |A| > 1, |B| > 1. andΔX = {(x,x) : x∈Xn\(A∪B)}. |
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