Geodesic
Epimorphisms between finite MV-algebras
Mathematica Bohemica, Tome 142 (2017) no. 4, pp. 345-355
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MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Łukasiewicz propositional calculus. Recently, algebraic theory of MV-algebras has been intensively studied. Wajsberg algebras are just a reformulation of Chang MV-algebras where implication is used instead of disjunction. Using these equivalence, in this paper we provide conditions for the existence of an epimorphism between two finite MV-algebras $A$ and $B$. Specifically, we define the mv-functions with domain in the ordered set of prime elements of $B$ and with range in the ordered set of prime elements of $A$, and prove that every epimorphism from $A$ to $B$ can be uniquely constructed from an mv-function.
MV-algebras were introduced by Chang to prove the completeness of the infinite-valued Łukasiewicz propositional calculus. Recently, algebraic theory of MV-algebras has been intensively studied. Wajsberg algebras are just a reformulation of Chang MV-algebras where implication is used instead of disjunction. Using these equivalence, in this paper we provide conditions for the existence of an epimorphism between two finite MV-algebras $A$ and $B$. Specifically, we define the mv-functions with domain in the ordered set of prime elements of $B$ and with range in the ordered set of prime elements of $A$, and prove that every epimorphism from $A$ to $B$ can be uniquely constructed from an mv-function.
DOI : 10.21136/MB.2017.0077-14
Classification : 06D35, 08A35
Keywords: MV-algebras; mv-function; epimorphism 
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Figallo, Aldo V.; Lattanzi, Marina B. Epimorphisms between finite MV-algebras. Mathematica Bohemica, Tome 142 (2017) no. 4, pp. 345-355. doi: 10.21136/MB.2017.0077-14

[1] Abad, M., Figallo, A.: On Łukasiewicz Homomorphisms. Facultad de Filosofía, Humanidades y Artes, Universidad Nacional de San Juan (1992).

[2] Berman, J., Blok, W. J.: Free Łukasiewicz and hoop residuation algebras. Stud. Log. 77 (2004), 153-180. | DOI | MR | JFM

[3] Boicescu, V., Filipoiu, A., Georgescu, G., Rudeanu, S.: Łukasiewicz-Moisil Algebras. Annals of Discrete Mathematics 49. North-Holland, Amsterdam (1991). | DOI | MR | JFM

[4] Chang, C. C.: Algebraic analysis of many valued logics. Trans. Am. Math. Soc. 88 (1958), 467-490. | DOI | MR | JFM

[5] Chang,, C. C.: A new proof of the completeness of Łukasiewicz axioms. Trans. Am. Math. Soc. 93 (1959), 74-80. | DOI | MR | JFM

[6] Cignoli, R., D'Ottaviano, I. M. L., Mundici, D.: Algebraic Foundations of Many-Valued Reasoning. Trends in Logic-Studia Logica Library 7. Kluwer Academic Publishers, Dordrecht (2000). | DOI | MR | JFM

[7] Cignoli, R., Dubuc, E. J., Mundici, D.: Extending Stone duality to multisets and locally finite $\rm MV$-algebras. J. Pure Appl. Algebra 189 (2004), 37-59. | DOI | MR | JFM

[8] Cignoli, R., Marra, V.: Stone duality for real-valued multisets. Forum Math. 24 (2012), 1317-1331. | DOI | MR | JFM

[9] Figallo, A. V.: Algebras implicativas de Łukasiewicz $(n+1)$-valuadas con diversas operaciones adicionales. Tesis Doctoral. Univ. Nac. del Sur (1990).

[10] Font, J. M., Rodríguez, A. J., Torrens, A.: Wajsberg algebras. Stochastica 8 (1984), 5-31. | MR | JFM

[11] Komori, Y.: Super-Łukasiewicz implicational logics. Nagoya Math. J. 72 (1978), 127-133. | DOI | MR | JFM

[12] Komori, Y.: Super Łukasiewicz propositional logics. Nagoya Math. J. 84 (1981), 119-133. | DOI | MR | JFM

[13] Łukasiewicz, J.: On three-valued logics. Ruch filozoficzny 5 (1920), 169-171 Polish.

[14] Łukasiewicz, J., Tarski, A.: Untersuchungen über den Aussagenkalkül. C. R. Soc. Sc. Varsovie 23 (1930), 30-50. | JFM

[15] Martínez, N. G.: The Priestley duality for Wajsberg algebras. Stud. Log. 49 (1990), 31-46. | DOI | MR | JFM

[16] Monteiro, L. F.: Number of epimorphisms between finite Łukasiewicz algebras. Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 49(97) (2006), 177-187. | MR | JFM

[17] Rodríguez, A. J.: Un studio algebraico de los cálculos proposicionales de Łukasiewicz. Ph. Doc. Diss. Universitat de Barcelona (1980).

[18] Rodríguez, A. J., Torrens, A., Verdú, V.: Łukasiewicz logic and Wajsberg algebras. Bull. Sect. Log., Pol. Acad. Sci. 19 (1990), 51-55. | MR | JFM

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