+ + = + +

All of the above structures are Dedekind complete (every nonempty bounded subset has a least upper bound). Narens and Luce [1976], Cohen and Narens [1979] and Narens [1981] investigate conditions under which positive concatenation structures are extendable to Dedekind complete ones. The arguments are much more complicated and subtle than those familiar from the associative case, and several results are established concerning what sort of measurement-theoretic properties are inherited by such Dedekind completions. The interested reader should consult the above papers. Throughout the rest of this part we consider only the Dedekind complete case with a dense automorphism group.