Continuous semigroup structures on , cancellative semigroups and bands

摘要

Up to isomorphism there is only one continuous group operation on , as Acz,l (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 modulo isomorphism. On the other hand, there are many continuous non-cancellative semigroup structures on . We classify all ordered continuous bands (idempotent semigroups) on . There are exactly eight (five) distinct ordered continuous band structures on modulo isomorphism (and anti-isomorphism).