A Generalization of Monadic n-Valued Łukasiewicz Algebras

Abstract

${{\mathcal {M}} L}^{m}_n$ of monadic m-generalized Łukasiewicz algebras of order n (or $M L^{m}_n$-algebras), namely a generalization of monadic n-valued Łukasiewicz algebras. In this article, we determine the congruences and we characterized the subdirectly irreducible $M L^{m}_n$-algebras. From this last result we proved that ${{\mathcal {M}} L}^{m}_n$ is a discriminator variety and as a consequence we characterized the principal congruences. In the last part of this paper we find an immersion of these algebras in a functional algebra and we proved that in the finite case they are isomorphic. This last result allows to show a new functional representation for monadic n-valued Łukasiewicz algebras. Finally, we define the notions of $M L^{m}_n$-algebra of fractions and maximal algebra of fractions and we prove the existence of a maximal $ML^{m}_n$-algebra of fractions for an $ML^{m}_n$-algebra.