  Here we initiate an investigation into the class mLMn{\times}m of monadic
n{\times}m-valued Lukasiewicz-Moisil algebras (or mLMn{\times}m-algebras),
namely n{\times}m-valued Lukasiewicz-Moisil algebras endowed with a unary
operation called existential quantifier. These algebras constitute a
generalization of monadic n-valued Lukasiewicz-Moisil algebras. In this
article, the relationship between existential quantifiers and special
subalgebras of mLMn{\times}m-algebras is shown. Besides, the congruences on
these algebras are determined and subdirectly irreducible algebras are
characterized. From this last result it is proved that mLMn{\times}m is a
discriminator variety and as a consequence, the principal congruences are
characterized. Furthermore, the number of congruences of finite
mLMn{\times}m-algebras is computed. In addition, a topological duality for
mLMn{\times}m-algebras is described and a characterization of
mLMn{\times}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.
