In this paper, a construction method on a bounded lattice obtained from a given t-norm on a subinterval of the bounded lattice is presented. The supremum distributivity of the constructed t-norm by the mentioned method is investigated under some special conditions. It is shown by an example that the extended t-norm on L from the t-norm on a subinterval of L need not be a supremum-distributive t-norm. Moreover, some relationships between the mentioned construction method and the other construction methods in the literature are presented.
S-measures are special fuzzy measures decomposable with respect to some fixed t-conorm S. We investigate the relationship of S-measures with some distinguished properties of fuzzy measures, such as subadditivity, submodularity, belief, etc. We show, for example, that each SP-measure is a plausibility measure, and that each S-measure is submodular whenever S is 1-Lipschitz.
In this paper, an equivalence on the class of uninorms on a bounded lattice is discussed. Some relationships between the equivalence classes of uninorms and the equivalence classes of their underlying t-norms and t-conorms are presented. Also, a characterization for the sets admitting some incomparability w.r.t. the U-partial order is given.