Residual operations of monotone binary operations over complete lattices

Abstract Various generalizations of triangular norms have been proposed, they are monotone binary operations satisfying commutativity or associativity or neutral property. Residual operations derived from these operations over complete lattices have been studied extensively. In this paper, residual operations of monotone binary operations over complete lattices are considered. Characterizations for these residual operations being implications or coimplications are given. Infinite distributivity, commutativity, associativity, neutral elements and annihilators of monotone binary operations and their residual operations are characterized. Finally, properties of residual operations derived from left (right) semi-uninorms, left (right) uninorms and (pseudo) uninorms are studied, and constructions of left (right) semi-uninorms, left (right) uninorms and (pseudo) uninorms through residuation are given.

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