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On \(\Gamma\)-semigroup. III. (English) Zbl 0652.20061

[Part II, cf. ibid. 79, No.6, 331-335 (1987; Zbl 0641.20042).]
Let M be a \(\Gamma\)-semigroup. Let E be the set of idempoen with the Vietoris topology is assigned to an object X in the category of compact Hausdorff spaces and continuous mappings, and the morphism \(f^{(1)}:X^{(1)}\to Y^{(1)}\), where \(f^{(1)}\) is the extension of f to \(X^{(1)}\) is assigned to a morphism f:X\(\to Y\). This assignment is a covariant functor and when applied to the union mapping \(u:X^{(2)}\to X^{(1)}\), by iteration, an inverse sequence \(\{X^{(n)},u^{(n)}\}\) is obtained and \(X^{(\omega)}\) denotes its limit space.
If \(C_{-1}\), \(C_ 0\), \(C_ 1=C\) are the empty set, the one-point space, the Cantor set, respectively, then for \(n\geq 2\), by interpolation of a copy of \(C_{n-3}+C_{n-2}\) in each removed interval of C, the space \(C_ n\) is obtained.
We prove: If X is compact metric and zero-dimensional, then \(X^{(\omega)}\) is one of the spaces \(C_ 0\), \(C_ 1\), \(C_ 0+C_ 1\), \(C_ 1+C_ 2\) and \(C_ 4\). A necessary and sufficient condition for an element of an inverse limit space to be an isolated point is also given.

MSC:

20M99 Semigroups
20N99 Other generalizations of groups
20M10 General structure theory for semigroups
20M15 Mappings of semigroups
20M50 Connections of semigroups with homological algebra and category theory

Citations:

Zbl 0641.20042
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