 The Countability Indices of Certain Transformation SemigroupsK. D. Magill
A chapter in Semigroups and Their Applications, 1987, pp 91-97 from Springer Abstract: Abstract The countability index C(S) of a semigroup S is defined to be the smallest positive integer N, if it exists, such that every countable subset of S is contained in a subsemigroup with N generators. If no such integer exists, we define C(S) = ∞. If S is noncommutative, C(S) ≥ 2. The rather surprising fact is that it is not all that rare for a full endomorphism semigroup to have count-ability index two. It has been known for quite a while, for example, that C(S(IN)) = 2 where S(IN) is the semigroup of all continuous selfmaps of the Euclidean N-cell, IN. In this paper, we recount the history of the subject and we discuss some recent results concerning the countability index of the endomorphism semigroup of a vector space. We conclude by showing that given any semigroup T, there exists a compact Hausdorff space X such that T can be embedded in S(X) and C(S(X)) = 2 where S(X) is the semigroup of all continuous selfmaps of X. Keywords: Continuous Extension; Compact Hausdorff Space; Semi Group; Countable Subset; Topological Semigroup (search for similar items in EconPapers) There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-009-3839-7_12 Ordering information: This item can be ordered from DOI: 10.1007/978-94-009-3839-7_12 Access Statistics for this chapter More chapters in Springer Books from Springer |
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