On monadic MV-algebras. (English) Zbl 1052.06010

The variety MMV of monadic MV-algebras was introduced by Rutledge as an algebraic model for a predicate calculus of Łukasiewicz infinite-valued logic with a single variable. This variety is a generalization of Halmos’s monadic Boolean algebras. For every monadic MV-algebra \(A\) it was proved by Rutledge that the set \(E\) of existential elements of \(A\) is a relatively complete subalgebra of \(A\). The authors prove that the existential part actually satisfies a stronger property, called \(m\)-relative completeness. As a consequence, they establish a one-one correspondence between monadic MV-algebras and pairs of MV-algebras \((A,E)\) where \(E\) is an \(m\)-relatively completmented subalgebra of \(A\). They study the ideal structure of MMV, and prove that MMV is congruence distributive and has the congruence extension property. Using Rutledge subdirect representation theorem, they prove that a finite monadic MV-algebra \(A\) with totally ordered existential part \(E\) is isomorphic to a product of totally ordered MV-algebras. The final section deals with a special subclass of MMV, called free cyclic. For background on MV-algebras see [R. L. O. Cignoli, I. M. L. D’Ottaviano, and D. Mundici, Algebraic foundations of many-valued reasoning. Kluwer Academic Publishers, Dordrecht (2000; Zbl 0937.06009)]. Rutledge’s results appeared in his PhD Thesis [A preliminary investigation of the infinitely many-valued predicate calculus, Cornell University (1959)].


06D35 MV-algebras
03G25 Other algebras related to logic
08B10 Congruence modularity, congruence distributivity
08B05 Equational logic, Mal’tsev conditions


Zbl 0937.06009
Full Text: DOI


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