Sentence examples for a minimal element in from inspiring English sources

Since x ∗ is a minimal element in ( V ( G ) ε 0, ≺ ∗ ), we get x = x ∗.

Then, for each x ∈ X, S ( x ) has a minimal element in S ( x ), where S ( x ) = { y ∈ X : y ⊑ x }.

Similarly, we can prove that F has a minimal element in (D_{1}), noting that F is bounded below by the zero matrix.

Since, we can replace by in the above assertions to obtain a minimal element in the sense, where ia defined here as (2.3).

By applying an argument similar to that in the proof of [9, Theorem ], it follows that is compact (the topology on is that of weak pointwise convergence) and there is a minimal element in the following sense: (x2a).

By applying an argument similar to that in the proof of [9, Theorem ] (see also the proof of [7, Lemma ]), it follows that is compact and there is a minimal element in the following sense: (3.4).

Thus, in view of the Kuratowski-Zorn lemma, we get a minimal element (A_) in Γ.

Using Zorn's Lemma, we obtain a minimal element K in F. Then K be a minimal subset of C with respect to being nonempty, closed, convex and satisfying the property.

Therefore, there exists a minimal element (e(f)) in Φ, which is an essential set in (F'(f,q)).

Thus, x ⪯ η x n for all n ≥ 0. By Lemma 3.2, S ( x 0 ) has a minimal element x ¯ in S ( x 0 ).

Since any descending chain in (Omega ) has a non-empty lower bound given by pointwise intersection, by Zorn's Lemma, we can find a minimal element (omega _0 in Omega ).