In this paper, we study uni-nullnorms and null-uninorms on bounded lattices, which are generalizations of uninorms, nullnorms, t-norms and t-conorms. We construct two uni-nullnorms and two null-uninorms on a bounded lattice [Formula: see text]. We determine the smallest-greatest uni-nullnorms with 2-neutral element [Formula: see text] and the smallest-greatest null-uninorms with 2-neutral element [Formula: see text] by using these construction methods.

In this paper, the concept of [Formula: see text]-compatible congruence and its related properties are studied. We also focus on homomorphism theorems for uninorms. Decomposition of uninorms is investigated via factor congruences, at first. Subsequently, we investigate decomposition of conjunctive and disjunctive uninorms, respectively. After all, we determine necessary and sufficient conditions for decomposition of some uninorms that are neither conjunctive nor disjunctive. In addition to all these, we shed light on the application of our results with the examples.

Having in mind that ordinal sum construction of triangular norms (triangular conorms) may not work on bounded lattices, in general, we propose a modification of ordinal sums of t-norms (t-conorms) resulting to a t-norm (t-conorm) on an arbitrary bounded lattice. In particular, our method can be applied to define connectives for fuzzy sets type 2, interval-valued fuzzy sets, intuitionistic fuzzy sets, etc. Some illustrative examples are added, considering the interval lattice L([0, 1]), the intuitionistic lattice LI, and the diamond lattice.