Sentence examples for be a preordered set from inspiring English sources

Let ( P, ≽ ) be a preordered set.

Let ( P, ≽ P ) be a preordered set.

Definition 2.3 Let ( X, ⪯ ) be a preordered set and d be a metric on X.

Lemma 3.3 Let ( S i, ≽ i ) be a preordered set for every i = 1, 2, …, n.

Let ( X, ⪯ ) be a preordered set, and let T : X → X be a mapping.

Let ( X, ≼ ) be a preordered set endowed with a G ∗ -metric G and T : X → X be a given mapping.

Then ≽ S is a preorder on S; and hence ( S, ≽ S ) is a preordered set.

Let R 0 + = [ 0, ∞ ). Definition 1 We will say that ≼ is a partial preorder on X (or ( X, ≼ ) is a preordered set or ( X, ≼ ) is a partially preordered space) if the following properties hold.

Remark 6 Obviously, Ψ ⊂ Φ. Definition 7 We will say that ≼ is a partial preorder on X (or ( X, ≼ ) is a preordered set or ( X, ≼ ) is a partially preordered space) if the following properties hold.

It is clear that ≽ k is a preorder relation on R k. Hence ( R k, ≽ k ) is a preordered set.

Recall that if ( X, ⪯ ) is a preordered set and T : X → X is such that for x, y ∈ X, with x ⪯ y implies T x ⪯ T y, then the mapping T is said to be nondecreasing.