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Ordered power associative groupoids

Author: Desmond A. Robbie
Journal: Proc. Amer. Math. Soc. 31 (1972), 285-290
MathSciNet review: 0285662
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Abstract: Compact, connected, totally ordered, (Hausdorff) topological groupoids, with restrictions on their sets of idempotents and with varying degrees of power associativity assumed, are examined. The paper evolves from the author's example of such a groupoid which has only two idempotents (a zero for least element, and an identity for greatest element), a compact neighborhood of the greatest element consisting of power associative elements, and which is not isomorphic to either the real thread or the nil thread. Another example given has a zero for least element, an idempotent for greatest element, and no other idempotents, and has a compact neighborhood of the greatest element consisting of an associative subgroupoid in which all products are equal to the greatest element. Theorems are given which show that these examples, and one other, in some sense, exhaust the possibilities.

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

  • [1] J. Aczél, Quasigroups, nets and nomograms, Advances in Math. 1 (1965), 383-450. MR 33 #1395. MR 0193174 (33:1395)
  • [2] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York; Addison-Wesley, Reading, Mass., 1963, pp. 181-188. MR 30 #2090. MR 0171864 (30:2090)
  • [3] J. G. Hocking and G. S. Young, Topology, Addison-Wesley, Reading, Mass., 1961. MR 23 #A2857. MR 0125557 (23:A2857)
  • [4] K. H. Hofmann, Zur mathematischen Theorie des Messens, Rozprawy Mat. 32 (1963), 32 pp. MR 33 #3275. MR 0195070 (33:3275)
  • [5] K. H. Hofmann and P. S. Mostert, Elements of compact semigroups, Merrill, Columbus, Ohio, 1966. MR 35 #285. MR 0209387 (35:285)
  • [6] 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)
  • [7] P. S. Mostert, Comments on a paper of Warne, Mimeographed sheets, Tulane University, New Orleans, La., 1963.
  • [8] D. A. Robbie, Some theorems on binary topological algebras, Dissertation, University of Florida, Gainesville, Fla., 1970.
  • [9] K. N. Sigmon, Cancellative medial means are arithmetic, Duke Math. J. 38 (1970), 143-146. MR 0274644 (43:407)
  • [10] R. J. Warne, Connected ordered topological groupoids with idempotent endpoints, Publ. Math. Debrecen 8 (1961), 143-146. MR 24 #A538. MR 0130678 (24:A538)
  • [11] R. L. Wilder, Topology of manifolds, Amer. Math. Soc. Colloq. Publ., vol. 32, Amer. Math. Soc., Providence, R.I., 1949. MR 10, 614. MR 0029491 (10:614c)

Additional Information

Keywords: Ordered groupoid, power associative, sect, cancellative, real thread, nil thread, isomorphism
Article copyright: © Copyright 1972 American Mathematical Society

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