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SPECTRAL DUALITIES OF MV-ALGEBRAS

  • Choe, Tae-Ho (Department of Mathematics & Statistics McMaster University) ;
  • Kim, Eun-Sup (Department of Mathematics College of Natural Sciences Kyungpook National University) ;
  • Park, Young-Soo (Department of Mathematics College of Natural Sciences Kyungpook National University)
  • Published : 2005.11.01

Abstract

Hong and Nel in [8] obtained a number of spectral dualities between a cartesian closed topological category X and a category of algebras of suitable type in X in accordance with the original formalism of Porst and Wischnewsky[12]. In this paper, there arises a dual adjointness S $\vdash$ C between the category X = Lim of limit spaces and that A of MV-algebras in X. We firstly show that the spectral duality: $S(A)^{op}{\simeq}C(X^{op})$ holds for the dualizing object K = I = [0,1] or K = 2 = {0, 1}. Secondly, we study a duality between the category of Tychonoff spaces and the category of semi-simple MV-algebras. Furthermore, it is shown that for any $X\;\in\;Lim\;(X\;{\neq}\;{\emptyset})\;C(X,\;I)$ is densely embedded into a cube $I^/H/$, where H is a set.

Keywords

MV-algebra;spectral duality;limit space;topological Boolean algebra;semi-simple MV-algebra;Tychonoff space;zero-dimensional space

References

  1. J. Adamek and H. Herrlich, Abstract and Concrete Categories, John Wiley & Sons, Inc., 1990
  2. L. P. Belluce, Semisimple algebras of infinite-valued logic and bold fuzzy set theory, Canad. J. Math. 38 (1986), 1356-1379 https://doi.org/10.4153/CJM-1986-069-0
  3. L. P. Belluce, $\alpha$-complete MV-algebras, Non-classi. log and their appl. to fuzzy subsets, Linz. 1992, 7-21
  4. C. C. Chang, Algebraic analysis of many valued logics, Trans. Amer. Math Soc. 88 (1958), 467-490 https://doi.org/10.2307/1993227
  5. T. H. Choe, A dual adjointness on partially ordered topological spaces, J. Pure Appl. Algebra 68 (1990), 87-93 https://doi.org/10.1016/0022-4049(90)90135-5
  6. T. H. Choe, E. S. Kim, and Y. S. Park, Representations of semi-simple MV - algebra, Kyungpook Math. J. 45 (2005), to appear
  7. L. Gillman and M. Jerison, Rings of Continuous Functions, Van Nostrand Princeton, NJ., 1960
  8. S. S. Hong and L. D. Nel, Duality theorems for algebras in convenient categories, Math. Z. 166 (1979), 131-136 https://doi.org/10.1007/BF01214038
  9. A. Di Nola and S. Sessa, On MV -algebras of continuous functions, Kluw, Acad. Pub. D. 1995, 23-32
  10. C. S. Hoo, Topological MV -algebras, Topology Appl. 81 (1997), 103-121 https://doi.org/10.1016/S0166-8641(97)00027-8
  11. D. Mundici, Interpretation of AFC-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63 https://doi.org/10.1016/0022-1236(86)90015-7
  12. H. E. Porst and M. B. Wischnewsky, Every topological category is convenient for Gelfand-Naimark duality, Manuscripta Math. 25 (1978), 169-204 https://doi.org/10.1007/BF01168608

Cited by

  1. An extension of Stone Duality to fuzzy topologies and MV-algebras vol.303, 2016, https://doi.org/10.1016/j.fss.2015.11.011