Abstract
Given an-dimensional right vector spaceV over a division ring\(\mathbb{K}\) we denote byS the semigroup of the endomorphisms ofV and designate this semigroup as alinear semigroup. First we prove that every automorphism ofS can be written asT→fTf −1, wheref∶V→V is a semilinear homeomorphism. Furthermore, we show that every isomorphism between maximal compact subsemigroups ofS is also of this type.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Clifford, A. H., and G. B. Preston, “The Algebraic Theory of Semigroups, Vol.I,” American Mathematical Society, Providence, 1961.
Cohn, P. M., “Algebra, Vol. III, Second Edition”, John Wiley & Sons, Chichester, New York, Brisbane, Toronto, Singapore, 1982.
Diedonné, J.,On the Automorphisms of the Classical Groups, Memoirs of the American Mathematical Society2 (1951).
Hofmann, K. H., and P. S. Mostert, “Elements of Compact Semigroups,” Charles E. Merrill Books, Columbus, 1966.
Shneperman, L. B.,On maximal compact subsemigroups of the endomorphism semigroup of an n-dimensional complex vector space, Semigroup Forum47 (1993), 197–209.
van Rooij, A. C. M., “Non-Archimedean Functional Analysis”, Marcel Dekker, Inc., New York, Basel, 1978.
Warner, S., “Topological Fields,” North-Holland Mathematics Studies, Amsterdam, New York, Oxford, Tokyo, 1989.
Weil, A., “Basic Number Theory,” Springer Verlag, Berlin, Heidelberg, New York, 1967.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Schwachhöfer, M. Automorphisms of linear semigroups over division rings. Semigroup Forum 53, 330–345 (1996). https://doi.org/10.1007/BF02574148
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02574148