Continuous semigroup structures on $\mathbb {R}$, cancellative semigroups and bands

Abstract

Up to isomorphism there is only one continuous group operation on $\mathbb {R}$, as Aczél (Bull Soc Math Fr 76: 59–64, 1949) showed in 1949. In this paper we show that there are exactly three distinct continuous (weakly) cancellative semigroup structures on $\mathbb {R}$ modulo isomorphism. On the other hand, there are many continuous non-cancellative semigroup structures on $\mathbb {R}$. We classify all ordered continuous bands (idempotent semigroups) on $\mathbb {R}$. There are exactly eight (five) distinct ordered continuous band structures on $\mathbb {R}$ modulo isomorphism (and anti-isomorphism).