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ENDS FOR MONOIDS AND SEMIGROUPS

Part of: Semigroups

Published online by Cambridge University Press:  01 August 2009

DAVID A. JACKSON  and
VESNA KILIBARDA
DAVID A. JACKSON*
Affiliation:
Department of Mathematics, Saint Louis University, St. Louis, MO 63103, USA (email: jacksoda@slu.edu)
VESNA KILIBARDA
Affiliation:
Department of Mathematics, Indiana University Northwest, Gary, IN 46408, USA (email: vkilibar@iun.edu)
*
✉For correspondence; e-mail: jacksoda@slu.edu
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Abstract

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We give a graph-theoretic definition for the number of ends of Cayley digraphs for finitely generated semigroups and monoids. For semigroups and monoids, left Cayley digraphs can be very different from right Cayley digraphs. In either case, the number of ends for the Cayley digraph does not depend upon which finite set of generators is used for the semigroup or monoid. For natural numbers m and n, we exhibit finitely generated monoids for which the left Cayley digraphs have m ends while the right Cayley digraphs have n ends. For direct products and for many other semidirect products of a pair of finitely generated infinite monoids, the right Cayley digraph of the semidirect product has only one end. A finitely generated subsemigroup of a free semigroup has either one end or else has infinitely many ends.

Keywords

endsmonoid and semigroupCayley digraphs

MSC classification

Secondary: 20M99: None of the above, but in this section
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 87 , Issue 1 , August 2009 , pp. 101 - 127
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2009

References

[1]Clifford, A. H. and Preston, G. B., The Algebraic Theory of Semigroups, Mathematical Surveys, 7 (American Mathematical Society, Providence, RI, 1961).Google Scholar
[2]Cohen, D. E., Groups of Cohomological Dimension One, Lecture Notes in Mathematics, 245 (Springer, Berlin, 1972).CrossRefGoogle Scholar
[3]Dunwoody, M. J., ‘The ends of finitely generated groups’, J. Algebra 12 (1969), 339–344.CrossRefGoogle Scholar
[4]Higgins, P. M., Techniques of Semigroup Theory (Oxford University Press, Oxford, 1992).CrossRefGoogle Scholar
[5]Howie, J. M., An Introduction to Semigroup Theory (Academic Press, London, 1976).Google Scholar
[6]Lyndon, R. C. and Schupp, P. E., Combinatorial Group Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, 89 (Springer, Berlin, 1977).Google Scholar
[7]Magnus, W., Karrass, A. and Solitar, D., Combinatorial Group Theory, Second Revised Edition (Dover Publications, New York, 1976).Google Scholar
[8]Petrich, M., Inverse Semigroups (Wiley, New York, 1984).Google Scholar
[9]Petrich, M. and Reilly, N. R., Completely Regular Semigroups (Wiley, New York, 1999).Google Scholar
[10]Schupp, P. E., ‘Groups and graphs’, Math. Intelligencer 1 (1979), 205–214.CrossRefGoogle Scholar
[11]Stallings, J., ‘On torsion-free groups with infinitely many ends’, Ann. of Math. 88 (1968), 312–334.CrossRefGoogle Scholar
[12]Stallings, J., Group Theory and Three-dimensional Manifolds, Yale Monographs, 4 (Yale University Press, New Haven, CT, 1971).Google Scholar