Sentence examples for incomparable elements from inspiring English sources

Now, let us construct a family of t-norms which are not distributive over each other with the help of incomparable elements in a bounded lattice.

Also, we constructed a family of t-norms which are not distributive over each other with the help of incomparable elements in a bounded lattice.

Finally, we construct a family of t-norms which are not distributive over each other with the help of incomparable elements in a bounded lattice.

In the present paper, we introduce an equivalence on the class of t-norms on ( [ 0, 1 ], ≤, 0, 1 ) based on the equality of the sets of all incomparable elements with respect to ⪯ T. The paper is organized as follows.

Define a relation ∼ on the class of all t-norms on ( [ 0, 1 ], ≤, 0, 1 ) by T 1 ∼ T 2 if and only if the set of all incomparable elements with respect to the T 1 -partial order is equal to the set of all incomparable elements with respect to the T 2 -partial order, that is, T 1 ∼ T 2 : ⇔ K T 1 = K T 2. Proposition 3.1 The relation ∼ given in Definition  3.1 is an equivalence relation.

We will use the notation K T to denote the set of all incomparable elements with respect to ⪯ T. Let L be a lattice and let T be any t-norm on L. In [10], a partial order for a t-norm T on L was defined.

Another betweenness interpretation enhancing the order-theoretic one used in [12] and [5] arises when the partial ordering (langle X,le rangle ) is a tree; i.e., the ordering is downwardly directed, but no two order-incomparable elements have a common upper bound.

On the other hand, if (ain A) and (a) is a cut point of the arc (A,) then (le _a) is a partial ordering which is not a tree ordering: the two noncut points of (A) are (le _a -incomparable, but any element of (S) is a common (le _a -incomparablefor them.

As shown in Figure 1 and Figure 2, when IΓ consists of two elements w1 and w2, a set of vectors incomparable with IΓ is given by the rectangle V. Let Γ be a vector incomparable with IΓ, i.e. γ ∈ V.

Moreover, if λ 1 T u 0 = u 0 holds with λ 1 = ( r ( T ) ) − 1, then for an arbitrary non-zero u ∈ K ( u ≠ k u 0 ) the elements u and λ 1 T u are incomparable.

The main obvious obstacles included: choice of elements in the overlapping parts of different models; incomparable sets of parameters; the lack of experimental data, as well as inability to use the data obtained for different cell lines or in different ways (e.g. single-cell or cell culture measurements); inability to make accurate predictions.