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$n$-valued Łukasiewicz-Moisil algebra; monadic $n$-valued Łukasiewicz-Moisil algebra; congruence; subdirectly irreducible algebra; discriminator variety; Priestley space
Here we initiate an investigation into the class $\boldsymbol m\boldsymbol L\boldsymbol M_{\boldsymbol n\boldsymbol \times \boldsymbol m}$ of monadic $n\times m$-valued Łukasiewicz-Moisil algebras (or $mLM_{n \times m}$-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 $\boldsymbol m\boldsymbol L\boldsymbol M_{\boldsymbol n\boldsymbol \times \boldsymbol m}$ is a discriminator variety and as a consequence, the principal congruences are characterized. Furthermore, the number of congruences of finite $mLM_{n \times m}$-algebras is computed. In addition, a topological duality for $mLM_{n \times m}$-algebras is described and a characterization of $mLM_{n \times m}$-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.
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