A unified way to studies of t-seminorms, t-semiconorms and semi-uninorms on a complete lattice in terms of behaviour operations

Abstract

We present a unified framework involved to the extensions of a t-seminorm, a t-semiconorm and a semi-uninorm in the lattice-valued setting through the use of the particular uplift and downlift of a binary operation linked by a behaviour operation. We also investigate the preservations of the left- and right-conjunctive of semi-uninorms as well as the left- and right-disjunctive of semi-uninorms under the extension approach studied in this paper. Moreover, for the extensions of t-seminorms, t-semiconorms and semi-uninorms, we provide their corresponding characterizations that depend on behaviour operations in terms of connected points. Finally, we summarize the dual results with respect to a unified way to investigations of t-seminorms, t-semiconorms and semi-uninorms.