Connections between MVn algebras and n-valued Lukasiewicz-Moisil algebras — II Original Research Article

I introduce in the MVn algebras of Revaz Grigolia two chains of unary operations, which are the key in establishing many connections between these algebras and n-valued Lukasiewicz-Moisil algebras (LMn algebras for short). The study has three parts. It is self-contained as much as possible. 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 and the class of ⊕-proper LMn algebras are introduced and studied. ⊕-proper LMn algebras coincide (can be identified) with Cignoliʹs proper n-valued Lukasiewicz algebras. MVn algebras coincide with good LMn algebras and with ⊕-proper LMn algebras (n⩾2). I 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 as LM3 algebras; the MVn algebras help us to see that.