Some functional representation theorems for monadic n-valued Luk asiewicz algebras (qLkn-algebras, for short) are given. Bearing in mind some of the results established by G. Georgescu and C. Vraciu (Algebre Boole monadice si algebre L ukasiewicz monadice, Studii Cercet. Mat. 23 (1971), 1027–1048) and P. Halmos (Algebraic Logic, Chelsea, New York, 1962), two functional representation theorems for qLkn-algebras are obtained. Besides, rich qLkn-algebras are introduced and characterized. In addition, a third theorem for these algebras is presented and the relationship between the three theorems is shown.
A topological duality for monadic n-valued Luk asiewicz algebras introduced by M. Abad (Abad, M.: Estructuras cíclica y monádica de un álgebra de L ukasiewicz nvalente. 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 of the associated space is also obtained. This allowed us to describe subdirectly irreducible algebras in this variety, arriving by a different method at the results established by Abad.