We can use the concept of a directed path to define a partial ordering relation between nodes within a directed graph.
Let X be a real ordered Banach space with a norm ∥ ⋅ ∥, zero θ, and a partial ordering relation ≤ defined by the normal cone P, and a normal constant N of P [4].
The partial ordering relation ≤ is defined in the following way: for every (a, b in M), (a le b) if and only if (a(x) le b(x)) for all (x in X).
The phone connections in the word-based phone lattice are broken, and the phone arcs are reorganized by their partial ordering relations.
Step 4. ∀C ∈ X, update partial ordering relations: Set C ≼ Cnew, if C ≼ C 1 * or C ≼ C 2 * ; set Cnew ≼ C, if C 1 * ≼ C or C 2 * ≼ C. Step 5. ∀(C1, C2) ∈ X × X, update partial ordering relations: Set C1 ≼ C2, if C 1 ≼ C 1 * and C 2 * ≼ C 2, or C 1 ≼ C 2 * and C 1 * ≼ C 2. Step 6. Set X = X ∪ { C new } { C 1 *, C 2 * }.
Note that if ( X, ⪯ ) is a partially ordered set, then we can endow X 3 with the following partial order relation: ( x, y, z ) ⪯ ( u, v, w ) ⟺ x ⪯ u, y ⪰ v, z ⪯ w. for all ( x, y, z ), ( u, v, w ) ∈ X 3 (see [26]).
Note that if ( X, ⪯ ) is a partially ordered set, then we can endow X × X with the following partial order relation: ( x, y ) ⪯ ( u, v ) ⟺ x ⪯ u, y ⪰ v. for all ( x, y ), ( u, v ) ∈ X × X [3].
Note that if (X,≼) is a partially ordered set, then we endow the product space X×X with the following partial order relation: for ( x, y ), ( u, v ) ∈ X × X, ( u, v ) ≼ ( x, y ) ⇔ x ≼ u, y ≽ v. Open image in new window.
Let be a partial order relation on.
Let (ll ) be a partial order relation on (mathbb R ).
Notice that ⪯ is a partial order relation on Z.
