If ( X, ≤ ) is a partially ordered set, the mapping F is said to have the mixed monotone property if x 1, x 2 ∈ X, x 1 ≤ x 2 ⇒ F ( x 1, y ) ≤ F ( x 2, y ), ∀ y ∈ X. and y 1, y 2 ∈ X, y 1 ≤ y 2 ⇒ F ( x, y 1 ) ≥ F ( x, y 2 ), ∀ x ∈ X.
Let ( x, ⪯ ) be a partially ordered set, the partial order ⪯2 for the product set X × X defined in the following way, for all ( x, y ), ( u, v ) ∈ X × X : ( x, y ) ⪯ 2 ( u, v ) ⇒ H ( x, y ) ⪯ H ( u, v ) and H ( v, u ) ⪯ H ( y, x ), where H X × X → → X is one-one.
Let ( X, ≤ ) be a partially ordered set, the subset E ⊂ X is said to be a totally ordered subset if either x ≤ y or y ≤ x holds for all x, y ∈ E. We say the elements x and y are comparable if either x ≤ y or y ≤ x holds.
Let ( x, ⪯ ) be a partially ordered set, the partial order ⪯3 for the product set X × X × X defined in the following way: for all ( x, y, z ), ( u, v, w ) ∈ X × X × X, ( x, y, z ) ⪯ 3 ( u, v, w ) if and only if G ( x, y, z ) ⪯ G ( u, v, w ), G ( v, w, u ) ⪯ G ( y, z, x ) and G ( w, u, v ) ⪯ G ( z, x, y ), where G X × X × X → → X is one-one.
Let (V,≼) be a totally ordered set where the set V={v0,…1,v,v m } is the set of forwarders.
Throughout this paper, ( X, ≤ ) denotes a partially ordered set with the partial order ≤.
Then ( X, ≤ ) is a partially ordered set with the natural ordering of real numbers.
Then ( X, ⪯ ) is a partially ordered set under the natural ordering of real numbers.
Throughout this article, (X, ≼) denotes a partially ordered set with the partial order ≼.
Example 2.2 Let X = R. Then ( X, ≤ ) is a partially ordered set with the natural ordering of real numbers.
The ordered set of polar lines allows the use of the planar straightness problem solution in the roundness problem.
