Sentence examples for be a preordered space from inspiring English sources

Let ( X, ⪯ ) be a preordered space, and F : X × X → X and g : X → X be two mappings.

Theorem 5.2 Let ( X, ≼ ) be a preordered space and let d : X × X → R and T, g : X → X be three mappings verifying (a - e).

Theorem 5.5 Let ( X, ≼ ) be a preordered space and let d : X × X → R and T, g : X → X be three mappings verifying the following properties.

Theorem 5.3 Let ( X, ≼ ) be a preordered space and let d : X × X → R and T, g : X → X be three mappings which fulfil conditions (a - h).

Now, we define the concept of weakly related mappings on preordered spaces as follows: Let ( X, ⪯ ) be a preordered space, and F : X × X → X and g : X → X be two mappings.

Remark 5.3 As we shall show in the proofs, the mapping d could only be considered on the set Ω = { ( x, y ) ∈ X 2 : x ≼ y }, that is, we will only use d | Ω : Ω → R. In this case, the previous remark shows that, as usual, d ⊆ [ 0, ∞ [. Theorem 5.1 Let ( X, ≼ ) be a preordered space and let T, g : X → X be two mappings verifying (a -(c).

From now on, ( X, ≼ ) will always denote a preordered space.

Let ( X, ≼ ) be a partially preordered space.

An preordered G-metric space is a triple ((X,G,preccurlyeq)) where ((X,G)) is a G-metric space and ≼ is a preordered on X.

A reflexive and transitive relation on X is a preordered on X.

Definition 16 Let ( X, G ) be a G n ∗ -metric space and let ≼ be a preorder on X.