J. Korean Math. Soc. 2005; 42(6): 1111-1120
Printed November 1, 2005
Copyright © The Korean Mathematical Society.
Tae Ho Choe, Eun Sup Kim, and Young Soo Park
McMaster University, Kyungpook National University, Kyungpook National University
Hong and Nel in [8] obtained a number of spectral dualities between a cartesian closed topological category ${\bf X}$ and a category of algebras of suitable type in ${\bf X}$ 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 ${\bf X}={\mathcal L}im$ of limit spaces and that ${\bf A}$ of $MV$-algebras in ${\bf X}.$ We firstly show that the spectral duality: $S({\bf A})^{op}\simeq C({\bf X}^{op})$ 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 $X\in{\mathcal L}im \, (X\neq\emptyset)$ $C(X, I)$ 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
MSC numbers: 03B50, 06D35
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