Some relations between left (right) semi-uninorms and coimplications on a complete lattice
Keming Tang, Zhudeng Wang · Systems Science & Control Engineering · 2015
Uninorms are important generalizations of triangular norms and conorms, with the neutral elements lying anywhere in the unit interval, left (right) semi-uninorms are non-commutative and non-associative extensions of uninorms, and coimplications are extensions of the Boolean coimplication. In this paper, we study the relationships between left (right) semi-uninorms and coimplications on a complete lattice. We first discuss the residual coimplicators of left and right semi-uninorms and show that the right (left) residual coimplicator of a disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorm is a right infinitely ∨-distributive coimplication which satisfies the neutrality principle. Then, we investigate the left and right semi-uninorms induced by a coimplication and demonstrate that the operations induced by right infinitely ∨-distributive coimplications, which satisfy the order property or neutrality principle, are left (right) infinitely ∧-distributive left (right) semi-uninorms or right (left) semi-uninorms. Finally, we prove that the meet-semilattice of all disjunctive right (left) infinitely ∧-distributive left (right) semi-uninorms is order-reversing isomorphic to the join-semilattice of all right infinitely ∨-distributive coimplications that satisfy the neutrality principle.