Topological Additively Representable Semigroups

We analyze the structure of nontrivial totally ordered connected topological semigroups S,q,Q., and prove that they are homeomorphic, algebraically iso- morphic, and isotonic to one of the following continua: a, `., q, F., wa, `.,q,F., aG0. y`, b.,q,F., y`, bx,q,F., bF0. or R,q,F., 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.