Generalizing the migrativity of continuous t-norms
The paper studies the migrativity property for continuous t-norms. In Section 1, two definitions which establish when a binary operation \(T:[0, 1]^2 \rightarrow [0, 1]\) is \(\alpha\)-migrative and \(\alpha\)-migrative with respect to a fixed t-norm \(T_0\) are recalled. Section 2 briefly recalls the definition of a t-norm and its representation as an ordinal sum. In the next section, continuous t-norms \(T\) which satisfy the relation \(T(\lambda_1x, \lambda_2y)=\lambda_3T(x,y)\) for given \(\lambda_i \in (0, 1]\), \(i=1,2,3\), with \(\lambda_1\geq \lambda_2 >\lambda_3\), are defined. In Theorem 3.2, a necessary and sufficient condition to satisfy that relation is proved. Other conditions are given by Corollaries 3.1 and 3.2 for additional conditions imposed for \(\lambda_1\) and \(\lambda_2\). Section 4 studies other cases: \(\lambda_1>\lambda_2=\lambda_3\) and \(\lambda_1=\lambda_2=\lambda_3\). In further papers, the author aims to investigate other binary operations and to extend the product from the relation investigated in this paper to other continuous t-norms.
scientific article; zbMATH DE number 6112225