Semigroup Forum (2017) 95:159–178
Continuous Archimedean semigroups on real intervals
· Sin-Ei Takahasi
Received: 23 July 2015 / Accepted: 28 April 2017 / Published online: 12 June 2017
© Springer Science+Business Media New York 2017
Abstract We study continuous Archimedean semigroups on real intervals. We give
a complete description of them using real functions called ample functions. We give
a criterion for two continuous Archimedean semigroups to be isomorphic in terms
of their ample functions. Using this characterization we construct uncountably many
non-isomorphic continuous Archimedean semigroups on a real interval.
Keywords Continuous semigroup · Topological semigroup · Continuous group ·
Topological group · Ordered semigroup · Archimedean semigroup · Standard function
The space R of real numbers is a topological group with the ordinary addition +.As
shown by Aczél  in 1949, the addition is a unique continuous group operation on
R up to isomorphism. Later, Craigen and Pales  generalized the results as follows:
a cancellative continuous semigroup on R is isomorphic to an additive semigroup of
a real open interval. In our previous paper  we give a complete classiﬁcation of
continuous ordered bands (all elements of them are idempotents) on R.
A continuous semigroup on R is Archimedean, if it has no idempotent. In this paper
we give a complete characterization of continuous Archimedean semigroups on a real
Communicated by Jimmie D. Lawson.
The authors were partially supported by JSPS Kakenhi Grand (C)-25400120.
Department of Information Science, Toho University, Funabashi 274-8510, Japan