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In this paper, we investigate several classes of binary operations within the preordered framework. First, we give the definition of quasi-t-subnorm on preordered sets. Subsequently, we research two classes of quasi-t-subnorms by means of Adjunctions and Left Galois Connections respectively. Meanwhile, we introduce new preorders differing from the inclusion order on powersets. Importantly, quasi-t-subnorms can be defined on powersets with preorders. Then we get a class of extensive binary operations via Adjunctions between preordered sets and the preordered sets consisting of powersets, which generalizes the previous results. The relationship between related binary operations is explored at last. [ABSTRACT FROM AUTHOR]

In this paper, we focus on the construction of non-direct product triangular norms on product lattices. First, we give one class of non-direct product triangular norms on more general product lattices, which generalizes the previous results. Subsequently, we provide another class of non-direct product triangular norms on product lattices, which differ from others. Our work gives an alternative answer to an open problem proposed by Jenei and De Baets. [ABSTRACT FROM AUTHOR]

In this paper, we present several necessary and sufficient conditions to ensure an ordinal sum of t-norms is still a t-norm on bounded lattices. For a fixed or an arbitrary bounded sublattice of a bounded lattice, we provide a direct characterization theorem, respectively. Meanwhile, we extend these results to the ordinal sum for a family of pairwise disjoint bounded sublattices. Furthermore, we investigate ordinal sum of t-norms on product lattices and reveal that there exist ordinal sum t-norms on non-trivial bounded sublattices of product lattices, which is different from the previous results of ordinal sum of t-norms defined on subintervals. [ABSTRACT FROM AUTHOR]