Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Semigroups over trees


Author: M. W. Mislove
Journal: Trans. Amer. Math. Soc. 195 (1974), 383-400
MSC: Primary 22A15
MathSciNet review: 0352321
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A semigroup over a tree is a compact semigroup S such that $ \mathcal{H}$ is a congruence on S and $ S/\mathcal{H}$ is an abelian tree with idempotent endpoints. Each such semigroup is characterized as being constructible from cylindrical subsemigroups of S and the tree $ S/\mathcal{H}$ in a manner similar to the construction of the hormos. Indeed, the hormos is shown to be a particular example of the construction given herein when $ S/\mathcal{H}$ is an I-semigroup. Several results about semigroups whose underlying space is a tree are also established as lemmata for the main results.


References [Enhancements On Off] (What's this?)

  • [1] K. W. Folley, Semigroups, Proc. Sympos., Semigroups, Wayne State University, Detroit, Mich. (June 27-29, 1968), Academic Press, New York, 1969. MR 41 #3637.
  • [2] K. H. Hofmann and M. W. Mislove, The centralizing theorem for left normal groups of units in compact monoiods, Semigroup Forum 3 (1971), 31-42. MR 0311830 (47:392)
  • [3] K. H. Hofmann and P. S. Mostert, Elements of compact semigroups, Merrill, Columbus, Ohio, 1966. MR 35 #285. MR 0209387 (35:285)
  • [4] R. P. Hunter, On the semigroup structure of continua, Trans. Amer. Math. Soc. 93 (1959), 356-368. MR 22 #82. MR 0109194 (22:82)
  • [5] J. L. Kelley, General topology, Van Nostrand, Princeton, N. J., 1955. MR 16, 1136. MR 0070144 (16:1136c)
  • [6] R. J. Koch, Threads in compact semigroups, Math. Z. 86 (1964), 321-316. MR 30 #1499. MR 0171268 (30:1499)
  • [7] R. J. Koch and I. S. Krule, Weak cutpoint ordering on hereditarily unicoherent continua, Proc. Amer. Math. Soc. 11 (1960), 679-681. MR 22 #11356. MR 0120606 (22:11356)
  • [8] R. J. Koch and A. D. Wallace, Maximal ideals in compact semigroups, Duke Math. J. 21 (1954), 681-685. MR 16, 112. MR 0063381 (16:112e)
  • [9] M. W. Mislove, Four problems about compact semigroups, Doctoral Dissertation, University of Tennessee, 1969.
  • [10] L. Nachbin, Sur les espaces topologiques ordonnés, C. R. Acad. Sci. Paris 226 (1948), 381-382; Sur les espaces uniformisables ordonnés, C. R. Acad. Sci. Paris 226 (1948), 547; Sur les espaces ordonnés, C. R. Acad. Sci. Paris 226 (1948), 774-775; English transl. in Topology and order, Van Nostrand Math. Studies, Van Nostrand, Princeton, N. J., 1965. MR 9, 367; 9, 455; 36 #2125. MR 0023516 (9:367b)
  • [11] R. C. Phillips, Interval clans with non-degenerate kernel, Proc. Amer. Math. Soc. 14 (1963), 396-400. MR 26 #5547. MR 0148038 (26:5547)
  • [12] L. E. Ward, Jr., On dendritic sets, Duke Math. J. 25 (1958), 505-513. MR 20 #4818. MR 0098357 (20:4818)
  • [13] -, Mobs, trees, and fixed points, Proc. Amer. Math. Soc. 8 (1957), 798-804. MR 20 #3516. MR 0097036 (20:3516)
  • [14] -, A note on dendrites and trees, Proc. Amer. Math. Soc. 5 (1954), 992-994. MR 17, 180. MR 0071759 (17:180c)
  • [15] M. W. Mislove, The existence of Irr(X), Trans. Amer. Math. Soc. 175 (1973), 123-140. MR 0325839 (48:4185)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 22A15

Retrieve articles in all journals with MSC: 22A15


Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0352321-8
PII: S 0002-9947(1974)0352321-8
Keywords: Compact semigroup, tree, semigroup over a tree, generalized collection, hormos, I-semigroup
Article copyright: © Copyright 1974 American Mathematical Society