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Here we initiate an investigation into the class $mL{M}_{n\times m}$ of monadic $n\times m$-valued Łukasiewicz-Moisil algebras (or $mL{M}_{n\times m}$-algebras), namely $n\times m$-valued Łukasiewicz-Moisil algebras endowed with a unary operation. These algebras constitute a generalization of monadic $n$-valued Łukasiewicz-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 $mL{M}_{n\times m}$ is a discriminator variety and as a consequence, the...

Some functional representation theorems for monadic $n$-valued Łukasiewicz algebras (qLk${}_n$-algebras, for short) are given. Bearing in mind some of the results established by G. Georgescu and C. Vraciu (Algebre Boole monadice si algebre Łukasiewicz monadice, Studii Cercet. Mat. 23 (1971), 1027–1048) and P. Halmos (Algebraic Logic, Chelsea, New York, 1962), two functional representation theorems for qLk${}_n$-algebras are obtained. Besides, rich qLk${}_n$-algebras are introduced and characterized. In addition,...