The Relations between Implications and Left (Right) Semi-Uninorms on a Complete Lattice

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 Lambda-distributive implication, which satisfies the order property, are left (right) infinitelyV-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 V-distributive left (right) semi-uninorms are right infinitely Lambda-distributive implications that satisfy the order property. Finally, we reveal the relationships between strict left (right)-conjunctive left (right) infinitely V-distributive left (right) semi-uninorms and right infinitely Lambda-distributive implications which satisfy the order property.