Implications, coimplications and left semi-uninorms on a complete lattice

In this paper, we firstly show that the N-dual operation of the right residual implication, which is induced by a left-conjunctive right arbitrary \vee-distributive left semi-uninorm, is the right residual coimplication induced by its N-dual operation. As a dual result, the N-dual operation of the right residual coimplication, which is induced by a left-disjunctive right arbitrary \wedge-distributive left semi-uninorm, is the right residual implication induced by its N-dual operation. Then, we demonstrate that the N-dual operations of the left semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left semi-uninorms. Finally, we reveal the relationships between conjunctive right arbitrary \vee-distributive left semi-uninorms induced by implications and disjunctive right arbitrary \wedge-distributive left semi-uninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.

Keywords:

Fuzzy connective , Implication , Coimplication , Left semi-uninorm , Neutrality principle