Filters and the weak almost periodic compactification of a discrete semigroup

Berglund, John F.; Hindman, Neil

doi:10.1090/S0002-9947-1984-0742410-4

Filters and the weak almost periodic compactification of a discrete semigroup
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by John F. Berglund and Neil Hindman PDF

Trans. Amer. Math. Soc. 284 (1984), 1-38 Request permission

Abstract:

The weak almost periodic compactification of a semigroup is a compact semitopological semigroup with certain universal properties relative to the original semigroup. It is not, in general, a topological compactification. In this paper an internal construction of the weak almost periodic compactification of a discrete semigroup is constructed as a space of filters, and it is shown that for discrete semigroups, the compactification is usually topological. Other results obtained on the way to the main one include descriptions of weak almost periodic functions on closed subsemigroups of topological groups, conditions for functions on the additive natural numbers or on the integers to be weak almost periodic, and an example to show that the weak almost periodic compactification of the natural numbers is not the closure of the natural numbers in the weak almost periodic compactification of the integers.

References

John F. Berglund, On extending almost periodic functions, Pacific J. Math. 33 (1970), 281–289. MR 412742
J. F. Berglund and K. H. Hofmann, Compact semitopological semigroups and weakly almost periodic functions, Lecture Notes in Mathematics, No. 42, Springer-Verlag, Berlin-New York, 1967. MR 0223483
J. F. Berglund, H. D. Junghenn, and P. Milnes, Compact right topological semigroups and generalizations of almost periodicity, Lecture Notes in Mathematics, vol. 663, Springer, Berlin, 1978. MR 513591
R. B. Burckel, Weakly almost periodic functions on semigroups, Gordon and Breach Science Publishers, New York-London-Paris, 1970. MR 0263963
W. W. Comfort, Ultrafilters: some old and some new results, Bull. Amer. Math. Soc. 83 (1977), no. 4, 417–455. MR 454893, DOI 10.1090/S0002-9904-1977-14316-4

Linear operators

Leonard Gillman and Meyer Jerison, Rings of continuous functions, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR 0116199

Abstract harmonic analysis

Neil Hindman, On density, translates, and pairwise sums of integers, J. Combin. Theory Ser. A 33 (1982), no. 2, 147–157. MR 677568, DOI 10.1016/0097-3165(82)90003-6
Neil Hindman, Sums equal to products in $\beta \textbf {N}$, Semigroup Forum 21 (1980), no. 2-3, 221–255. MR 590754, DOI 10.1007/BF02572552
Neil Hindman, Ultrafilters and combinatorial number theory, Number theory, Carbondale 1979 (Proc. Southern Illinois Conf., Southern Illinois Univ., Carbondale, Ill., 1979) Lecture Notes in Math., vol. 751, Springer, Berlin, 1979, pp. 119–184. MR 564927
Karl Heinrich Hofmann and Paul S. Mostert, Elements of compact semigroups, Charles E. Merrill Books, Inc., Columbus, Ohio, 1966. MR 0209387
Jimmie D. Lawson, Additional notes on continuity in semitopological semigroups, Semigroup Forum 12 (1976), no. 3, 265–280. MR 405381, DOI 10.1007/BF02195932
J. D. Lawson, Joint continuity in semitopological semigroups, Illinois J. Math. 18 (1974), 275–285. MR 335674
K. de Leeuw and I. Glicksberg, Applications of almost periodic compactifications, Acta Math. 105 (1961), 63–97. MR 131784, DOI 10.1007/BF02559535