Vector t-Norms With Applications

Some basic t-norms defined on [0, 1] are well known in many study areas and applications. However, more general extension of them into vector forms can be used in a lot of new decision-making realms. In this study, we first define preference vector on a linearly ordered set, which includes different special vectors that are mathematically equivalent but with different application backgrounds and practical meanings. As a reasoning rule, we provide a merging method for more than two preference vectors to be aggregated into a finally resultant preference vector. The merging method is based on a special vector value function, which can be seen as a reasonable counterpart of product t-norm, f(x, y) = xy (x and y belong to [0, 1]). In addition, we use bilinear frame to define corresponding four types of basic bilinear vector t-norms, which are just counterparts of basic t-norms from many aspects. Mathematically, preference vectors are related to decreasing matrices, and we find the general entry relation for the decreasing matrices, by which we prove that a decreasing matrix is commutative and closed under matrix multiplication. Thus, we finally present the definition of preference multiplication commutative monoid, which is equivalent to product bilinear vector t-norm. We also show illustrative examples and applications of some results.