Home Mathematics A relationship between the category of chain MV-algebras and a subcategory of abelian groups
Article
Licensed
Unlicensed Requires Authentication

A relationship between the category of chain MV-algebras and a subcategory of abelian groups

  • Homeira Pajoohesh EMAIL logo
Published/Copyright: August 4, 2021
Become an author with De Gruyter Brill
Author Information Explore this Subject
Mathematica Slovaca
From the journal Mathematica Slovaca Volume 71 Issue 4

Abstract

The category of MV-algebras is equivalent to the category of abelian lattice ordered groups with strong units. In this article we introduce the category of circled abelian groups and prove that the category of chain MV-algebras is isomorphic with the category of chain circled abelian groups. In the last section we show that the category of chain MV-algebras is a subcategory of abelian cyclically ordered groups.

Keywords: MV-algebra; cyclic group; cyclically ordered group
MSC 2010: 06D35; 20F60

  1. (Communicated by Roberto Giuntini)

References

[1] Belluce, L. P.: Semisimple algebras of infinite-valued logic and bold fuzzy set theory, Canadian J. Math 38 (1986), 1356–1379.10.4153/CJM-1986-069-0Search in Google Scholar

[2] Chang, C. C.: Algebraic analysis of many valued logics, Trans. Amer. Math. Soc. 88 (1958), 467–490.10.1090/S0002-9947-1958-0094302-9Search in Google Scholar

[3] Chang, C. C.: A new proof of the completeness of Łukasiewicz's axioms, Trans. Amer. Math. Soc. 93 (1959), 74–80.10.2307/1993423Search in Google Scholar

[4] Cignoli, R.—D'ottaviano, I. M. L.—Mundici, D.: Algebraic Foundations of Many-Valued Reasoning, Kluwer, Dordrecht, 2000.10.1007/978-94-015-9480-6Search in Google Scholar

[5] Di Nola, A.—Gerla, B.: Algebras of Łukasiewicz's Logic and their semiring reducts, Contemp. Math. 377 (2005), 131–144.10.1090/conm/377/06988Search in Google Scholar

[6] Ghorbani, S.—Torkzadeh, L.—Motamed, S.: (⊙, ⊕)-derivations and (⊖, ⊙)-derivations on MV-algebras, Iran. J. Math. Sci. Inform. 8 (2013), 75–90.Search in Google Scholar

[7] Gluschankof D.: Cyclic ordered groups and MV-algebras, Czechoslovak Math. J. 43 (1993), 249–263.10.21136/CMJ.1993.128391Search in Google Scholar

[8] Jakubíik, J.: Retracts of abelian cyclically ordered groups, Arch. Math. 25 (1989), 13–18.Search in Google Scholar

[9] Łukasiewicz, J.: O logice trójwartościowej, Ruch Filozoficzny 5 (1920), 169-171.Search in Google Scholar

[10] Mundici, D.: MV-ALGEBRAS, A short tutorial, Department of Mathematics University of Florence, 2007; http://www.matematica.uns.edu.ar/IXCongresoMonteiro/Comunicaciones/Mundicitutorial.pdf.Search in Google Scholar

[11] Mundici, D.: Interpretation of AF C*-algebras in Łukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15–63.10.1016/0022-1236(86)90015-7Search in Google Scholar

[12] Mundici, D.: The logic of Ulam's game with lies. Knowledge Belief and Strategic Interaction (C. Bicchieri and M.L. Dalla Chiara, eds.), Cambridge University Press, 1992, pp. 275–284.10.1017/CBO9780511983474.017Search in Google Scholar

[13] Mundici, D.: Ulam's game, Łukasiewicz logic and C*-algebras, Fund. Math. 18 (1993), 151–161.10.3233/FI-1993-182-405Search in Google Scholar

[14] Rieger, L. S.: O uspořádaných a cyklicky uspořádaných grupách I [On ordered and cyclically ordered groups I], Věstník Král. České spol. nauk, Třída matematicko-přírodovědná [Journal of the Royal Czech Society of Sciences, Mathematics and natural history] 6 (1946), 1–31 (in Czech).Search in Google Scholar

[15] Rieger, L. S.: O uspořádaných a cyklicky uspořádaných grupách II [On ordered and cyclically ordered groups II], Věstník Král. České spol. nauk, Třída matematicko-přírodovědná [Journal of the Royal Czech Society of Sciences, Mathematics and natural history] 1 (1947), 1–33 (in Czech).Search in Google Scholar

[16] Rieger, L. S.: O uspořádaných a cyklicky uspořádaných grupách III (On ordered and cyclically ordered groups III), Věstník Král. České spol. nauk, Třída matematicko-přírodovědná [Journal of the Royal Czech Society of Sciences, Mathematics and natural history] 1 (1948), 1–26 (in Czech).Search in Google Scholar

[17] Swierczowski, S.: On cyclically ordered groups, Fund. Math. 47 (1959), 161–166.10.4064/fm-47-2-161-166Search in Google Scholar

Received: 2020-04-26
Accepted: 2020-10-23
Published Online: 2021-08-04
Published in Print: 2021-08-26

© 2021 Mathematical Institute Slovak Academy of Sciences

Articles in the same Issue

  1. Regular Papers
  2. Quasi-decompositions and quasidirect products of Hilbert algebras
  3. Residuation in finite posets
  4. On a problem in the theory of polynomials
  5. Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
  6. Mapping properties of the Bergman projections on elementary Reinhardt domains
  7. Lemniscate-like constants and infinite series
  8. On the Oscillation of second order nonlinear neutral delay differential equations
  9. Oscillation theorems for certain second-order nonlinear retarded difference equations
  10. De la Vallée Poussin inequality for impulsive differential equations
  11. Hs-Boundedness of a class of Fourier Integral Operators
  12. Dynamical behavior of a P-dimensional system of nonlinear difference equations
  13. Some inequalities for exponentially convex functions on time scales
  14. Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
  15. The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
  16. A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
  17. Simulations of nonlinear parabolic PDEs with forcing function without linearization
  18. An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
  19. A relationship between the category of chain MV-algebras and a subcategory of abelian groups
https://doi.org/10.1515/ms-2021-0037
Keywords for this article
MV-algebra; cyclic group; cyclically ordered group

Articles in the same Issue

  1. Regular Papers
  2. Quasi-decompositions and quasidirect products of Hilbert algebras
  3. Residuation in finite posets
  4. On a problem in the theory of polynomials
  5. Fekete-Szegö problem for starlike functions connected with k-Fibonacci numbers
  6. Mapping properties of the Bergman projections on elementary Reinhardt domains
  7. Lemniscate-like constants and infinite series
  8. On the Oscillation of second order nonlinear neutral delay differential equations
  9. Oscillation theorems for certain second-order nonlinear retarded difference equations
  10. De la Vallée Poussin inequality for impulsive differential equations
  11. Hs-Boundedness of a class of Fourier Integral Operators
  12. Dynamical behavior of a P-dimensional system of nonlinear difference equations
  13. Some inequalities for exponentially convex functions on time scales
  14. Impact of different types of non linearity on the oscillatory behavior of higher order neutral difference equations
  15. The sine extended odd Fréchet-G family of distribution with applications to complete and censored data
  16. A new two-parameter lifetime distribution with flexible hazard rate function: Properties, applications and different method of estimations
  17. Simulations of nonlinear parabolic PDEs with forcing function without linearization
  18. An existence level for the residual sum of squares of the power-law regression with an unknown location parameter
  19. A relationship between the category of chain MV-algebras and a subcategory of abelian groups
Sign up now to receive a 20% welcome discount
Institutional Access
How does access work?
Have an idea on how to improve our website?
Please write us.
© 2025 De Gruyter Brill
Downloaded on 15.12.2025 from https://www.degruyterbrill.com/document/doi/10.1515/ms-2021-0037/html
Scroll to top button