Assume that
is a semi-order space. From the previous discussion, we have the following results.
Therefore, under the maximal (minimal) sense, the modification above is unique.
The idea of the construction of quotient semi-order that we offered is the following.
First, a right-order topology
is induced from
T. From
, a quotient topology
on
is induced. Finally, from
, a quotient (pseudo) semi-order on [
X] is induced. Now, the point is whether there exists another method for inducing quotient semi-order. If the method exists, the induced quotient semi-order is as the former, i.e., the uniqueness of quotient semi-order. Moreover, when
R and
T are incompatible, we show that the method
above cannot induce quotient semi-order on [
X]. Then, whether there is another method that can induce quotient semi-order on [
X] such that its projection is order-preserving. That is, the existence of quotient semi-order.
The following two propositions answer the above uniqueness and existence problems.