Topological additively representable semigroups

Abstract

We analyze the structure of nontrivial totally ordered connected topological semigroups (S,+,≲), and prove that they are homeomorphic, algebraically isomorphic, and isotonic to one of the following continua: ((a,∞),+,≤), ([a,∞),+,≤), (a≥0) ((-∞,b),+,≤), ((-∞,b],+,≤), (b≤0) or (R,+,≤), all considered as subsets of the totally ordered group of additive real numbers endowed with the usual (Euclidean) topology. We conclude with a general study about the continuity in the representation of topological totally ordered semigroups through additive real-valued order-preserving homomorphisms. © 1997 Academic Press.