SPECTRAL DUALITIES OF MV-ALGEBRAS

Title & Authors
SPECTRAL DUALITIES OF MV-ALGEBRAS
Choe, Tae-Ho; Kim, Eun-Sup; Park, Young-Soo;

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 $\small{\vdash}$ C between the category X = Lim of limit spaces and that A of MV-algebras in X. We firstly show that the spectral duality: $\small{S(A)^{op}{\simeq}C(X^{op})}$ 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 $\small{X\;\in\;Lim\;(X\;{\neq}\;{\emptyset})\;C(X,\;I)}$ is densely embedded into a cube $\small{I^/H/}$, where H is a set.ﾖ⨀
Keywords
MV-algebra;spectral duality;limit space;topological Boolean algebra;semi-simple MV-algebra;Tychonoff space;zero-dimensional space;
Language
English
Cited by
1.
An extension of Stone Duality to fuzzy topologies and MV-algebras, Fuzzy Sets and Systems, 2016, 303, 80
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

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

5.
T. H. Choe, A dual adjointness on partially ordered topological spaces, J. Pure Appl. Algebra 68 (1990), 87-93

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

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

11.
D. Mundici, Interpretation of AFC-algebras in Lukasiewicz sentential calculus, J. Funct. Anal. 65 (1986), 15-63

12.
H. E. Porst and M. B. Wischnewsky, Every topological category is convenient for Gelfand-Naimark duality, Manuscripta Math. 25 (1978), 169-204