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In 2000, Figallo and Sanza introduced [Formula: see text]-valued Łukasiewicz–Moisil algebras which are both a particular case of matrix Łukasiewicz algebras [W. Suchoń, Matrix Łukasiewicz Algebras, Rep. Math. Logic 4 (1975) 91–104] and a generalization of [Formula: see text]-valued Łukasiewicz–Moisil algebras. It is worth noting that unlike what happens in [Formula: see text]-valued Łukasiewicz–Moisil algebras, generally the De Morgan reducts of [Formula: see text]-valued Łukasiewicz–Moisil algebras are not Kleene algebras. Furthermore, in [C. Sanza, [Formula: see text]-valued Łukasiewicz algebras with negation, Rep. Math. Logic 40 (2006) 83–106] an important example which legitimated the study of this class of algebras is provided. In this paper, we continue the study of [Formula: see text]-valued Łukasiewicz–Moisil algebras (or [Formula: see text]-algebras). More precisely, we determine a new topological duality for these algebras. By means of this duality we characterize the congruences and specially the maximal congruences on [Formula: see text]-algebras. Then these characterizations allow us to assert that [Formula: see text]-algebras are semisimples and obtain a new description of subdirectly irreducible [Formula: see text]-algebras.

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In this paper, we continue the study of the Łukasiewicz residuation algebras of order $n$ with Moisil possibility operators (or $MC_n$-algebras) initiated by Figallo (1989, PhD Thesis, Universidad Nacional del Sur). More precisely, among other things, a method to determine the number of elements of the $MC_n$-algebra with a finite set of free generators is described. Applying this method, we find again the results obtained by Iturrioz and Monteiro (1966, Rev. Union Mat. Argent., 22, 146) and by Figallo (1990, Rep. Math. Logic, 24, 3–16) for the case of Tarski algebras and $I\varDelta _{3}$-algebras, respectively.