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The Countability Indices of Certain Transformation Semigroups

In: Semigroups and Their Applications

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  • K. D. Magill Jr.

    (SUNY at Buffalo, Mathematics Department)

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.

Suggested Citation

  • K. D. Magill Jr., 1987. "The Countability Indices of Certain Transformation Semigroups," Springer Books, in: Simon M. Goberstein & Peter M. Higgins (ed.), Semigroups and Their Applications, pages 91-97, Springer.
    • Handle: RePEc:spr:sprchp:978-94-009-3839-7_12
    DOI: 10.1007/978-94-009-3839-7_12
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