Semi-uninorms and implications on a complete lattice

In this paper, we study the relationships between left (right) semi-uninorms and implications on a complete lattice. We firstly discuss the residual operations of left and right semi-uninorms and show that the right (left) residual operator of a conjunctive right (left) ∨-distributive left (right) semi-uninorm is a right ∧-distributive implication that satisfies the neutrality principle. Then, we investigate the left and right semi-uninorms induced by an implication, give some conditions such that two operations induced by an implication constitute left or right semi-uninorms, and demonstrate that the operations induced by a right ∧-distributive implication, which satisfies the order property or neutrality principle, are left (right) ∨-distributive left (right) semi-uninorms or right (left) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) ∨-distributive left (right) semi-uninorms and right ∧-distributive implications which satisfy the neutrality principle.