We introduce two chains of unary operations in the MVn algebra of Revaz Grigolia; they will be used in establishing many connections between these algebras and n-valued Lukasiewicz-Moisil algebras (LMn algebras for short). The study has four parts. It is by and large self-contained. The main result of the first part is that MV4 algebras coincide with LM4 algebras. The larger class of ``relaxed''-MVn algebras is also introduced and studied. This class is related to the class of generalized LMn pre-algebras. The main results of the second part are that, for n 5, any MVn algebra is an LMn algebra and that the canonical MVn algebra can be identified with the canonical LMn algebra. In the third part, the class of good LMn algebras is introduced and studied and it is proved that MVn algebras coincide with good LMn algebras. In the present fourth part, the class of -proper LMn algebras is introduced and studied. -proper LMn algebras coincide (can be identified) with Cignoli's proper n-valued Lukasiewicz algebras. MVn algebras coincide with -proper LMn algebras (n 2). We also give the construction of an LM3(LM4) algebra from the odd (respectively even)-valued LMn algebra (n 5), which proves that LM4 algebras are as much important than LM3 algebras; MVn algebras help to see this point. 1 C.S.Calude and G.Stefanescu (eds.). Automata, Logic, and Computability. Special issue dedicated to Professor Sergiu Rudeanu Festschrift.