Quasi-uniformities on topological semigroups and completions
by
Hans-Peter A. Kunzi
Dept. Math. Appl. Math., University of Cape Town, Rondebosch 7701, South Africa
Coauthors: Josefa Marin (Polytechnical University of Valencia, Spain), Salvador Romaguera (Polytechnical University of Valencia, Spain)

By definitions analogous to those used in the case of topological groups one can define the left quasi-uniformity, the right quasi-uniformity and the two-sided quasi-uniformity on paratopological groups. In recent years the bicompletions (and even Doitchinov completions) of these quasi-uniformities for (regular) paratopological (Hausdorff) groups were studied in some detail and numerous interesting results were obtained.

So far the theory of these quasi-uniformities on arbitrary topological monoids is considerably more complicated and less satisfactory. Nevertheless for semigroups which are (left)-cancellable or which are locally totally bounded, various theorems similar to those known from the classical theory of topological groups could be established.

References

[1] H.-P. A. Kunzi, Quasi-uniform isomorphism groups, in Recent Progress in Function Spaces, G. Di Maio and L. Hola (eds.), Quaderni di matematica, Dip. Mat. Sec. Univ. Napoli, vol. 3, 1998, pp. 187-220.

[2] H.-P.A. Kunzi, Nonsymmetric distances and their associated topologies: about the origins of basic ideas in the area of asymmetric topology, in Handbook of the History of General Topology, vol. 3, C. Aull and R. Lowen (eds.), Kluwer, Dordrecht, pp. 853-968.

[3] H.-P. A. Kunzi, J. Marin and S. Romaguera, Quasi-uniformities on topological semigroups and bicompletion, Semigroup Forum 62 (2001), 403-422.

[4] H.-P. A. Kunzi, S. Romaguera and O.V. Sipacheva, The Doitchinov completion of a regular paratopological group, Serdica Math. J. 24 (1998), 73-88.

[5] J. Marin and S. Romaguera, A bitopological view of quasi-topological groups, Indian J. Pure Appl. Math. 27 (1996), 393-405.

[6] J. Marin and S. Romaguera, Bicompleting the left quasi-uniformity of a paratopological group, Arch. Math. (Basel) 70 (1998), 104-110.

Date received: April 14, 2002