Here we initiate an investigation into the class mLMn×m of monadic n × m-valued Lukasiewicz-Moisil algebras (or mLMn×m-algebras), namely n × m-valued Lukasiewicz-Moisil algebras endowed with a unary operation. These algebras constitute a generalization of monadic n-valued Lukasiewicz-Moisil algebras. In this article, the congruences on these algebras are determined and subdirectly irreducible algebras are characterized. From this last result it is proved that mLMn×m is a discriminator variety and as a consequence, the principal congruences are characterized. Furthermore, the number of congruences of finite mLMn×m-algebras is computed. In addition, a topological duality for mLMn×m-algebras is described and a characterization of mLMn×m-congruences in terms of special subsets of the associated space is shown. Moreover, the subsets which correspond to principal congruences are determined. Finally, some functional representation theorems for these algebras are given and the relationship between them is pointed out.
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.