Ordered power associative groupoids

Robbie, Desmond A.

doi:10.1090/S0002-9939-1972-0285662-7

Ordered power associative groupoids
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by Desmond A. Robbie

Proc. Amer. Math. Soc. 31 (1972), 285-290

DOI: https://doi.org/10.1090/S0002-9939-1972-0285662-7

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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

J. Aczél, Quasigroups, nets, and nomograms, Advances in Math. 1 (1965), no. fasc. 3, 383–450. MR 193174, DOI 10.1016/0001-8708(65)90042-3
L. Fuchs, Partially ordered algebraic systems, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR 0171864
John G. Hocking and Gail S. Young, Topology, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London, 1961. MR 0125557
K. H. Hofmann, Zur mathematischen Theorie des Messens, Rozprawy Mat. 32 (1963), 32 (German). MR 195070
Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
R. J. Koch and A. D. Wallace, Maximal ideals in compact semigroups, Duke Math. J. 21 (1954), 681–685. MR 63381

Comments on a paper of Warne

Some theorems on binary topological algebras

Kermit Sigmon, Cancellative medial means are arithmetic, Duke Math. J. 37 (1970), 439–445. MR 274644
R. J. Warne, Connected ordered topological groupoids with idempotent endpoints, Publ. Math. Debrecen 8 (1961), 143–146. MR 130678, DOI 10.5486/pmd.1961.8.1-2.13
Raymond Louis Wilder, Topology of Manifolds, American Mathematical Society Colloquium Publications, Vol. 32, American Mathematical Society, New York, N. Y., 1949. MR 0029491

Bibliographic Information

Journal: Proc. Amer. Math. Soc. 31 (1972), 285-290
DOI: https://doi.org/10.1090/S0002-9939-1972-0285662-7
MathSciNet review: 0285662