On unitary Cauchy filters on topological monoids

Boris G. Averbukh

Topological Algebra and its Applications (2013)

  • Volume: 1, page 46-59
  • ISSN: 2299-3231

Abstract

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For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.

How to cite

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Boris G. Averbukh. "On unitary Cauchy filters on topological monoids." Topological Algebra and its Applications 1 (2013): 46-59. <http://eudml.org/doc/266770>.

@article{BorisG2013,
abstract = {For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.},
author = {Boris G. Averbukh},
journal = {Topological Algebra and its Applications},
keywords = {topological monoid; Cauchy space; uniformity; Topological monoid; -net; unitary uniformity; unitary completeness},
language = {eng},
pages = {46-59},
title = {On unitary Cauchy filters on topological monoids},
url = {http://eudml.org/doc/266770},
volume = {1},
year = {2013},
}

TY - JOUR
AU - Boris G. Averbukh
TI - On unitary Cauchy filters on topological monoids
JO - Topological Algebra and its Applications
PY - 2013
VL - 1
SP - 46
EP - 59
AB - For Hausdorff topological monoids, the concept of a unitary Cauchy net is a generalization of the concept of a fundamental sequence of reals. We consider properties and applications of such nets and of corresponding filters and prove, in particular, that the underlying set of a given monoid, endowed with the family of such filters, forms a Cauchy space whose convergence structure defines a uniform topology. A commutative monoid endowed with the corresponding uniformity is uniform. A distant purpose of the paper is to transfer the classical concepts of a completeness and of a completion into the theory of topological monoids.
LA - eng
KW - topological monoid; Cauchy space; uniformity; Topological monoid; -net; unitary uniformity; unitary completeness
UR - http://eudml.org/doc/266770
ER -

References

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  1. [1] J.H. Carruth, J.A. Hildebrant, R.J. Koch, The theory of topological semigroups, Pure and Applied Mathematics, Marcel Dekker Inc., New York, 1983. Zbl0515.22003
  2. [2] R. Engelking, General topology. Rev. and compl. ed., Sigma Series in Pure Mathematics, 6., Berlin: Heldermann Verlag, 1989, viii + 529 pp., ISBN 3-88538-006-4, Zbl 0684.54001. 
  3. [3] H.H. Keller, Die Limes-Uniformisierbarkeit der Limesräume, Math. Ann., 1968, 176, 334-341. Zbl0155.50302
  4. [4] E. Lowen-Colebunders, Function Classes of Cauchy Continuous Maps, Pure and Applied Mathematics, Marcel Dekker Inc., New York, 1989. 
  5. [5] D. Marxen, Uniform semigroups, Math. Ann., 1973, 202, 27-36. Zbl0234.22003
  6. [6] J.F. Ramaley, O. Wyler, Cauchy spaces, http://repository.cmu.edu/math/ 97, 1968. Zbl0195.24402

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