A topological duality for monadic $n$-valued Łukasiewicz algebras introduced by M. Abad (Abad, M.: Estructuras cíclica y monádica de un álgebra de Łukasiewicz $n$-valente. Notas de Lógica Matemática 36. Instituto de Matemática. Universidad Nacional del Sur, 1988) is determined. When restricted to the category of $Q$-distributive lattices and $Q$-homomorphims, it coincides with the duality obtained by R. Cignoli in 1991. A new characterization of congruences by means of certain closed and involutive subsets...