Relations among implications, coimplications and left (right) semi-uninorms
Meixia Niu, Xiaoying Hao, Zhudeng Wang · Journal of Intelligent & Fuzzy Systems · 2015
Abstract In this paper, we further study implications, coimplications and left (right) semi-uninorms on a complete lattice. We firstly show that the N -dual operation of the right (left) residual implication, which is generated by a left (right)-conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorm, is the right (left) residual coimplication generated by its N -dual operation. As a dual result, the N -dual operation of the right (left) residual coimplication, which is generated by a left (right)-disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm, is the right (left) residual implication generated by its N -dual operation. Then, we demonstrate that the N -dual operations of the left (right) semi-uninorms induced by an implication and a coimplication, which satisfy the neutrality principle, are the left (right) semi-uninorms. Finally, we reveal the relationships between conjunctive right (left) infinitely ∨-distributive left (right) semi-uninorms induced by implications and disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms induced by coimplications, where both implications and coimplications satisfy the neutrality principle.