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