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.