Abstract
It was shown by the second named author that the equivalence classes of metrics on the two copies of a metric space X form an inverse semigroup M(X). Here we generalize this result to the uniform spaces. We define the composition of uniform structures on the two copies of a uniform space X and show that the equivalence classes of uniform structures are a semigroup U(X). We also give a characterization of idempotents in U(X). Our main result is that this semigroup is an inverse semigroup. We also compare this semigroup with Md(X) when (X,d) is a metric space.
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