Kybernetika 47 no. 5, 740-754, 2011

On the existence of a Haar measure in topological IP-loops

Beáta Stehlíková, Dagmar Markechová and Anna Tirpáková


In this paper, we give conditions ensuring the existence of a Haar measure in topological IP-loops.


quasigroup, topological IP-loop, Haar measure, content, uniform, space, left-invariant uniformity


28C10, 20N05


  1. J. Baez: The Octonions. Bull. Amer. Math. Soc. 39 (2002), 145-205.   CrossRef
  2. V. D. Belousov: Foundations of the Theory of Quasigroups and Loops (in Russian). Nauka, Moscow, 1967.   CrossRef
  3. R. H. Bruck: Some results in the theory of quasigroups. Trans. Amer. Math. Soc. 55 (1944), 19-52.   CrossRef
  4. R. H. Bruck: Contributions to the theory of loops. Trans. Amer. Math. Soc. 60 (1946), 245-354.   CrossRef
  5. A. Haar: Der Massbegriff in der Theorie der kontinuierlichen Gruppen. Ann. Math. 34 (1933), 147-169.   CrossRef
  6. P. Halmos: Measure Theory. Springer-Verlag New York Inc. 1974.   CrossRef
  7. S. N. Hudson: Transformation groups in the theory of topological loops. Proc. Amer. Math. Soc. 15 (1964), 872-877.   CrossRef
  8. S. N. Hudson: Topological loops with invariant uniformities. Trans. Amer. Math. Soc. 109 (1963), 1, 181-190.   CrossRef
  9. O. Chein, H. O. Pflugfelder and J. D. H. Smith: Quasigroups and Loops. Theory and Application. Heldermann Verlag, 1990.   CrossRef
  10. M. K. Kinyon, K. Kunen and J. D. Phillips: Every diassociative A-loop is Moufang. Proc. Amer. Math. Soc. 130 (2002), 619-624.   CrossRef
  11. D. Markechová, B. Stehlíková and A. Tirpáková: The uniqueness of a left Haar measure in topological IP-loops. (Submitted for publication).   CrossRef
  12. R. Moufang: Zur Struktur von alternativ Korpern. Math. Ann. 110 (1935), 416-430.   CrossRef
  13. P. T. Nagy and K. Strambach: Coverings of topological loops. J. Math. Sci. 137 (2006), 5, 5098-5116.   CrossRef
  14. P. T. Nagy and K. Strambach: Loops in Group Theory and Lie Theory. De Gruyter Expositions in Mathematics, Berlin - New York 2002.   CrossRef
  15. H. O. Pflugfelder: Quasigroups and Loops. Introduction. Heldermann Verlag, 1990.   CrossRef
  16. B. Riečan and T. Neubrunn: Integral, Measure and Ordering. Kluwer Academic Publishers 1997.   CrossRef
  17. Š. Schwarz: On the existence of invariant measures on certain types of bicompact semigroups (in Russian). Czechoslovak Math. J. 7 (1957), 82, 165-182.   CrossRef
  18. J. D. H. Smith: Quasigroup Representation Theory. Taylor \et Francis, 2006.   CrossRef
  19. A. Weil: Basic Number Theory. Academic Press, 1971.   CrossRef