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1. Introduction
It is well known that a general tolerance space can not admit usual algebraic structures [5]. Therefore in the study of difference geometry, the contravariant tensor bundles (for example, the tangent bundle) are lack of algebraic structures. However, we can obtain algebraic structure in the covariant bundle, that is, from the dual point of view[2].
In §2 and §3, some results of T. Poston are reviewed. In §4, we give the definitions of tensor product, exterior product, and obtain similar results as in differential geometry. In §5, we try to define difference forms by another method. In §6, we explain why the big fuzzy condition is necessary in the study of algebraic fuzzy. These give us some insight into the theory of difference geometry. Finally some algebraic results about the fuzzy on 2^x are obtained in §7. Especially we point out the possibility to extend these results to general Boolean algebras.
2. Basic definitions and results
A tolerance space, or fuzzy space (X,μ) is a set X with a symmetric reflexive relation μ□X×X, called the tolerance or fuzzy on X. If (x,y) εμ, then x is withen fuzzy of y, or indistinguishable from y. The set μ(x0={y|(x,y)εμ}□X is the fuzzy neighborhood of x in X.
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