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Semigroup structures for families of functions, I. Some Homomorphism Theorems

Published online by Cambridge University Press:  09 April 2009

Kenneth D. Magill Jr
Kenneth D. Magill Jr
Affiliation:
State University of New York at Buffalo
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This is the first of several papers which grew out of an attempt to provide C (X, Y), the family of all continuous functions mapping a topological space X into a topological space Y, with an algebraic structure. In the event Y has an algebraic structure with which the topological structure is compatible, pointwise operations can be defined on C (X, Y). Indeed, this has been done and has proved extremely fruitful, especially in the case of the ring C (X, R) of all continuous, real-valued functions defined on X [3]. Now, one can provide C(X, Y) with an algebraic structure even in the absence of an algebraic structure on Y. In fact, each continuous function from Y into X determines, in a natural way, a semigroup structure for C(X, Y). To see this, let ƒ be any continuous function from Y into X and for ƒ and g in C(X, Y), define ƒg by each x in X.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 7 , Issue 1 , February 1967 , pp. 81 - 94
Copyright
Copyright © Australian Mathematical Society 1967

References

[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Mathematical Surveys, Number 7, Amer. Math. Soc. 1961.Google Scholar
[2]Fuchs, L., Partially Ordered Algebraic Systems, Pergamon Press, Addison-Wesley, Reading, Mass. 1963.Google Scholar
[3]Gillman, L. and Jerison, M., Rings of Continuous Functions, D. Van Nostrand, New York, 1960.CrossRefGoogle Scholar
[4]Ljapin, E. S., semigroups, Vol. 3, Translations of Mathematical Monographs, Amer. Math. Soc., 1963.Google Scholar
[5]Ljapin, E. S., ‘Abstract Characterization of Certain Semigroups of Transformations’, Leningrad. Gos. Ped. Inst. Ucen. Zap. 103 (1955), 5–29 (Russian).Google Scholar
[6]Magill, K. D. Jr, ‘Semigroups of Continuous Functions’, Amer. Math. Monthly, 71 (1964), 984–988.CrossRefGoogle Scholar
[7]Magill, K. D. Jr, ‘Some Homomorphism Theorems For a Class of Semigroups’, Proc. London Math. Soc. (3) 15 (1965), 517–526.CrossRefGoogle Scholar
[8]Magill, K. D. Jr, ‘Semigroups of Functions on Topological Spaces’, Proc. London Math. Soc. (3) 16 (1966), 507–518.CrossRefGoogle Scholar
[9]Malcev, A. I., ‘Symmetric Groupoids’, Math. Sb. (N.S.) 31 (73) (1952), 136–151 (Russian).Google Scholar
[10]Schreier, I., ‘Über Abbildungen einer abstrakten Menge auf ihre Teilmenge’. Fund. Math. 28 (1936), 261–264.CrossRefGoogle Scholar