Existence of Additive and Continuous Utility Functions on Ordered Semigroups

We study the continuity of additive utility functions on (totally) ordered topological semigroups endowed with the order topology. The existence of continuous additive utilities is characterized through the semigroup being translation-invariant and super-Archimedean, and connectedness turns out to be a sufficient condition allowing to substitute the milder assumption of cancellativity to translation invariance. A classification result in the connected case follows.