Narrow Results

The α-migrativity is an important property of a binary operation, and many authors discussed this property for various operators. In this paper, the migrative property for nullnorms is investigated. All solutions of the migrativity equation for all possible combinations of nullnorms are analyzed and characterized.

Uninorms are important generalizations of triangular norms and conorms, with a neutral element lying anywhere in the unit interval, and left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms. In this paper, we study the relations between implications and left (right) semi-uninorms on a complete lattice. We firstly investigate the left (right) semi-uninorms induced by implications, give some conditions such that the operations induced by implications constitute left or right semi-uninorms, and demonstrate that the operations induced by a right infinitely ∧-distributive implication, which satisfies the order property, are left (right) infinitely ∨-distributive left (right) semi-uninorms. Then, we discuss the residual operations of left (right) semi-uninorms and show that left (right) residual operators of strict left (right)-conjunctive left (right) infinitely ∨-distributive left (right) semi-uninorms are right infinitely ∧-distributive implications that satisfy the order property. Finally, we reveal the relationships between strict left (right)-conjunctive left (right) infinitely ∨-distributive left (right) semi-uninorms and right infinitely ∧-distributive implications which satisfy the order property.

In this paper, we firstly introduce two new classes of fuzzy implications generated from one-variable functions, called $\text{(f, g,}\wedge \text{)}$- and $\text{(f, g,}\vee \text{)}$-implications, respectively. Then we give a series of necessary and sufficient conditions that these implications satisfy: left neutrality property, identity principle, ordering principle, law of contraposition, modus ponens and modus tollens, respectively. We also discuss the relations between $\text{(f, g,}\wedge \text{)}$- implication ($\text{(f, g,}\vee \text{)}$-implications, respectively) and other known classes of fuzzy implications.

This paper is mainly devoted to investigating the migrativity equations involving nullnorms and 2-uninorms. Depending on whether the absorbing elements of 2-uninorms and unllnorms are same or not, all solutions of the migrativity equations for all possible combinations of the three defined subclasses of 2-uninorms and nullnorms are analyzed and characterized respectively. And for such equations, there are new solutions which extend the known ones about the migrativity for uninorms and nullnorms.

In this paper, we show that the set of all left semi-uninorms and the set of all implications on a complete lattice are all complete lattices and obtain the formulas for calculating the smallest left semi-uninorm, the largest left semi-uninorm, the largest implication and the smallest implication generated by a binary operation.