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)].

### MSC:

 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:

### References:

 [1] H. Bass, Finite monadic algebras, Proc. Amer. Math. Soc. (1958) 258-268. · Zbl 0089.01904 [2] Belluce, L.P., Further results on infinite valued predicate logic, J. symbolic logic, 29, 69-78, (1964) · Zbl 0127.00801 [3] Belluce, L.P.; Chang, C.C., A weak completeness theorem for infinite valued first-order logic, J. symbolic logic, 28, 43-50, (1963) · Zbl 0121.01203 [4] Birkhoff, G., Lattice theory, (1967), Providence Rhode Island · Zbl 0126.03801 [5] Chang, C.C., Algebraic analysis of many-valued logics, Trans. amer. math. soc., 88, 467-490, (1958) · Zbl 0084.00704 [6] Di Nola, A.; Grigolia, R.; Panti, G., Finitely generated free MV-algebras and their automorphism groups, Stud. logica, 61, 65-78, (1998) · Zbl 0964.06010 [7] Georgescu, G.; Iurgulescu, A.; Leustean, I., Monadic and closure MV-algebras, Multi. val. logic, 3, 235-257, (1998) · Zbl 0920.06004 [8] Grätzer, G., Universal algebra, (1978), Springer New York [9] R. Grigolia, Algebraic analysis of Lukasiewicz-Tarski n-valued logical systems, Selected Papers on Lukasiewicz Sentential Calculi, Wroclaw, 1977, pp. 81-91. [10] Grigolia, R., Free algebras of non-classical logics, (1987), Metsniereba Press Tbilisi [11] L.S. Hay, An axiomatization of the infinitely many-valued calculus, M.S. Thesis, Cornell University, 1958. [12] Lukasiewicz, J.; Tarski, A., Unntersuchungen über den aussagenkalkul, Comptes rendus des seances de la societe des sciences et des lettres de varsovie, 23, cl iii, 30-50, (1930) · JFM 57.1319.01 [13] Mangani, P., On certain algebras related to many-valued logics, Boll. un. mat. ital., 4, 68-78, (1973) [14] McNaughton, R., A theorem about infinite-valued sentential logics, J. symbolic logic, 16, 1-13, (1951) · Zbl 0043.00901 [15] Mundici, D., Interpretation of $$AFC\^{}\{*\}$$-algebras in lukasiewicz sentential calculus, J. funct. anal., 65, 15-63, (1986) · Zbl 0597.46059 [16] J.D. Rutledge, A preliminary investigation of the infinitely many-valued predicate calculus, Ph.D. Thesis, Cornell University, 1959. [17] Scarpellini, B., Die nichtaxiomatisierbarkeit des unendlichwertigen prädikaten-kalkulus von lukasiewicz, J. symbolic logic, 27, 159-170, (1962) · Zbl 0112.24503 [18] Schwartz, D., Theorie der polyadischen MV-algebren endlicher ordnung, Math. nachr., 78, 131-138, (1977) · Zbl 0402.03054 [19] Schwartz, D., Polyadic MV-algebras, Zeit. f. math. logik und grundlagen d. math., 26, 561-564, (1980) · Zbl 0488.03035
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.