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In this construction <math>O_1 \subseteq S\!</math> is that portion of the original signs <math>S\!</math> for which higher order signs are created in the initial step of reflection, thereby being converted into <math>O_1 \subseteq O'.\!</math> The sign domain <math>S\!</math> is extended to a new sign domain <math>S'\!</math> by the addition of these higher order signs, namely, the set <math>S_1.\!</math> Using arch quotes, the mapping from <math>O_1\!</math> to <math>S_1\!</math> can be defined as follows:

In this construction O1 c S is that portion of the original signs S for which HO signs are created in the initial step of reflection, thereby being converted into O1 c O'.  The sign domain S is extended to a new sign domain S' by the addition of these HO signs, namely, the set S1.  Using the arch quotes (<...>), the mapping from O1 to S1 can be defined as follows:

Nom1 : O1 -> S1  such that   Nom1 : x -> <x>.

| <math>\operatorname{Nom}_1 : O_1 \to S_1 ~\text{such that}~ \operatorname{Nom}_1 : x \mapsto {}^{\langle} x {}^{\rangle}.\!</math>

Finally, the reflectively extended naming function Nom' : O'  > S' is defined as Nom' = Nom U Nom'.

Finally, the reflectively extended naming function Nom' : O'  > S' is defined as Nom' = Nom U Nom'.