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The Hope property and free products of semigroups

Part of: Semigroups

Published online by Cambridge University Press:  09 April 2009

P. R. Jones
P. R. Jones
Affiliation:
Mathematics DepartmentUniversity of Western Australia Nedlands, Western AustraliaAustralia6009
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Abstract

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The free product of two Hopfian groups (in the category of groups) need not be Hopfian. We prove, by elementary methods, that the free product of two simple Hopfian inverse semigroups is Hopfian. In particular the free product of any two Hopfian groups, in the category of inverse semigroups, is again Hopfian. In fact the same is true in the category of all semigroups.

MSC classification

Secondary: 20M05: Free semigroups, generators and relations, word problems
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 27 , Issue 3 , May 1979 , pp. 358 - 364
Copyright
Copyright © Australian Mathematical Society 1979

References

Bruck, R. H. (1958), A survey of binary systems (Ergebnisse der Mathematik, Heft, 20, Springer, Berlin).CrossRefGoogle Scholar
Clifford, A. H. and Preston, G. B. (1961), Algebraic theory of semigroups, Vol. 1, Math. Surveys 7 (American Math. Soc., Providence, R.I.).CrossRefGoogle Scholar
Clifford, A. H. and Preston, G. B. (1967), Algebraic theory of semigroups, Vol. II, Math. Surveys 7 (American Math. Soc., Providence, R.I.).CrossRefGoogle Scholar
Dey, I. M. S. and Neumann, H. (1970), ‘The Hopf property of free products’, Math. Z. 117, 325–339.CrossRefGoogle Scholar
Hirshon, R. (1975), ‘The intersection of the subgroups of finite index in some finitely presented groups’, Proc. Amer. Math. Soc. 53, 32–36.CrossRefGoogle Scholar
Howie, J. M. (1976), An introduction to semigroup theory (Academic Press, London).Google Scholar
Kočin, B. P. (1968), ‘The structure of inverse ideal-simple ω-semigroups’, Vestnik Leningrad Univ. Math. 23, 41–50 (in Russian).Google Scholar
McAlister, D. B. (1968), ‘A homomorphism theorem for semigroups’, J. London Math. Soc. 43, 355–366.CrossRefGoogle Scholar
McAlister, D. B. (1976), ‘Inverse semigroups generated by a pair of subgroups’, Proc. Roy. Soc. Edinburgh Sect. A 77A, 9–22.CrossRefGoogle Scholar
Munn, W. D. (1968), ‘Regular ω-semigroups’, Glasgow Math. J. 9, 46–66.CrossRefGoogle Scholar
Munn, W. D. and Reilly, N. R. (1966), ‘Congruences on a bisimple ω-semigroup’, Proc. Glasgow Math. Assoc. 7, 184–192.CrossRefGoogle Scholar
Newman, M. F. and Sichler, J. (1973), ‘Free products of Hopf groups’, Math. Z. 135, 69–72.CrossRefGoogle Scholar
Preston, G. B. (1973), ‘Inverse semigroups: some open questions’, Proc. Symposium on Inverse Semigroups and Their Generalizations, pp. 122–139 (Northern Illinois University).Google Scholar
Reilly, N. R. (1976), ‘Free inverse semigroups’, manuscript.Google Scholar