Square Roots and Continuity in Strictly Linearly Ordered Semigroups on   Real Intervals
  In this article we show that the semigroup operation of a strictly linearly ordered semigroup on a real interval is automatically continuous if each element of the semigroup admits a square root. Hence, by a result of Acz\'el, such a semigroup is isomorphic to an additive subsemigroup of the real numbers. 
Jochen Gl\"uck
math.CA math.FA