Deresiduums of implications on a complete lattice

Abstract

In this paper, we first discuss some properties of residuum and deresiduum of a binary operation on a complete lattice. Then, we investigate the relations between residuum and deresiduum. Finally, we demonstrate how each individual logical character of a right infinitely ∧-distributive implication, such as the law of importation, the weak exchangeability principle, the exchange principle, the contrapositive symmetry and the contraction law, can be translated into a property of its deresiduum. Moreover, we give some conditions under which deresiduum of a right infinitely ∧-distributive implication is, respectively, a left (right) uninorm, pseudo-uninorm and uninorm, and show that right infinitely ∧-distributive implications, which satisfy the weak exchangeability principle and the contrapositive symmetry, can be presented by commutative conjunctors and strong negations.