Decorated linear order types and the theory of concatenation

We study the interpretation of Grzegorczyk’s Theory of Concatenation TC in structures of decorated linear order types satisfying Grzegorczyk’s axioms. We show that TC is incomplete for this interpretation. What is more, the first order theory validated by this interpretation interprets arithmetical truth. We also show that every extension of TC has a model that is not isomorphic to a structure of decorated order types. We provide a positive result, to wit a construction that builds structures of decorated order types from models of a suitable concatenation theory. This construction has the property that if there is a representation of a certain kind, then the construction provides a representation of that kind.