Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-06-09T02:46:51.589Z Has data issue: false hasContentIssue false
  • English
  • Français

On minimally Free Algebras

Published online by Cambridge University Press:  20 November 2018

Paul Bankston  and
Richard Schutt
Paul Bankston
Affiliation:
Marquette University, Milwaukee, Wisconsin
Richard Schutt
Affiliation:
2546 North 7th Street, Milwaukee, Wisconsin
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For us an “algebra” is a finitary “universal algebra” in the sense of G. Birkhoff [9]. We are concerned in this paper with algebras whose endomorphisms are determined by small subsets. For example, an algebra A is rigid (in the strong sense) if the only endomorphism on A is the identity idA. In this case, the empty set determines the endomorphism set E(A). We place the property of rigidity at the bottom rung of a cardinal-indexed ladder of properties as follows. Given a cardinal number κ, an algebra A is minimally free over a set of cardinality κ (κ-free for short) if there is a subset X ⊂ A of cardinality κ such that every function f:X → A extends to a unique endomorphism ϕ ∊ E(A). (It is clear that A is rigid if and only if A is 0-free.) Members of X will be called counters; and we will be interested in how badly counters can fail to generate the algebra.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 37 , Issue 5 , 01 October 1985 , pp. 963 - 978
Copyright
Copyright © Canadian Mathematical Society 1985

References

1. Armstrong, J., On p-pure subgroups of the p-adic integers, in Topics in abelian groups (Scott Foresman, Chicago, 1963).Google Scholar
2. Banaschewski, B., On lattices of continuous functions, Quaestiones Mathematicae 6 (1983), 1–12.Google Scholar
3. Bankston, P., Corrigendum to ‘Some obstacles to duality in topological algebra’ Can. J. Math 57 (1985), 82–83.Google Scholar
4. Bowshell, R. A. and Schultz, P., Unital rings whose additive endomorphisms commute, Math. Ann. 228 (1911), 197–214.Google Scholar
5. Comfort, W. W. and Negrepontis, S., The theory of ultrafilters (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar
6. Comfort, W. W. and Retta, T., Generalized perfect maps and a theorem of I. Juhasz, (to appear in the Proceedings of the A.M.S. Special Session on Rings of Continuous Functions).Google Scholar
7. Fuchs, L., Infinite abelian groups, Vol. II (Academic Press, New York, 1973).Google Scholar
8. Gillman, L. and Jerison, M., Rings of continuous functions (Van Nostrand, Princeton, 1960).CrossRefGoogle Scholar
9. Jónsson, B., Topics in universal algebra (Springer-Verlag, Berlin, 1972).CrossRefGoogle Scholar
10.(private communication).Google Scholar
11. Juhasz, I., Cardinal functions in topology (Mathematisch Centrum, Amsterdam, 1975).Google Scholar
12. Křiž, I. and Pultr, A., Large κ-free algebras, Alg. Univ. (to appear).CrossRefGoogle Scholar
13. Kunen, K., Set theory (North Holland, Amsterdam, 1980).Google Scholar
14. Nelson, E., (private communication).Google Scholar
15. Petrich, M., Inverse semigroups, (manuscript).Google Scholar
16. Pultr, A., (private communication).Google Scholar
17. Schultz, P., The endomorphism ring of the additive group of a ring, J. Aust. Math. Soc. 15 (1973), 60–69.Google Scholar
18. Sikorski, R., Boolean aleebras (Springer-Verlag, Berlin, 1960).Google Scholar
19. Walker, R., The Stone-Čech compactification (Springer-Verlag, Berlin, 1974).CrossRefGoogle Scholar
20. Williams, S. W., More realcompact spaces (to appear in the Proceedings of the A.M.S. Special Session on Rings of Continuous Functions).Google Scholar