Let us now consider that $\sigma_1 \in (\mathfrak{S} \setminus \overline{\mathfrak{S}})$ and $\sigma_2, \sigma_3, \sigma_4 \in \overline{\mathfrak{S}}^{\text{pure}}$. Using properties (6.44) and $\sigma_1 \asymp \sigma_2, \sigma_2 \asymp \sigma_3, \sigma_3 \asymp \sigma_4, \sigma_1 \asymp \sigma_4$ and (6.50)(6.3), we note that we can always choose $\pi, \rho \in \overline{\mathfrak{S}}$ such that

$$ \pi, \rho \in \Theta_{\overline{\mathfrak{S}}} (\sigma_1), \quad \sigma_1 = \pi \llcorner_{\overline{\mathfrak{S}}} \rho \qquad (9.59) $$

$$ \sigma_4 \overline{\Box}_{\overline{\mathfrak{S}}} \rho, \quad \sigma_2 \overline{\Box}_{\overline{\mathfrak{S}}} \pi \qquad (9.60) $$

$$ . \qquad (9.61) $$

Let us now assume that there exists $\lambda \in \mathfrak{S}^{\mathfrak{S}}$ such that $\tau^U(\lambda, \sigma_1, \sigma_2)$ and $\tau^U(\lambda, \sigma_3, \sigma_4)$. We have then

$$ \lambda \overline{\Box}_{\overline{\mathfrak{S}}} \pi, \quad \lambda \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_3 \Box_{\overline{\mathfrak{S}}} \sigma_4). \qquad (9.62) $$

The consistency relation $\sigma_1 \asymp \sigma_3$ means that there exists $\varepsilon \in \Theta_{\overline{\mathfrak{S}}} (\sigma_1)$ such that $\varepsilon \overline{\Box}_{\overline{\mathfrak{S}}} \sigma_3$. The consistency relation $\sigma_2 \asymp \sigma_4$ contains nothing.

Once again, we insist on the fact that $\varepsilon \neq \pi$ (resp. $\varepsilon \neq \rho$) because this would mean $\sigma_1 \Box_{\overline{\mathfrak{S}}} \sigma_3 = \varepsilon = \pi = \sigma_1 \Box_{\overline{\mathfrak{S}}} \sigma_2$ (resp. $\sigma_1 \Box_{\overline{\mathfrak{S}}} \sigma_3 = \varepsilon = \rho = \sigma_1 \Box_{\overline{\mathfrak{S}}} \sigma_4$). These degenerate cases have been excluded by assumption.

Let us now choose $\chi := \sigma_1$ with $\kappa := \sigma_2$ in (3.26), and simultaneously $\chi := \sigma_1$ with $\kappa := \sigma_3$ in (3.26), and simultaneously $\chi := \sigma_1$ with $\kappa := \sigma_4$ in (3.26). We obtain

$$ \pi \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_2 \Box_{\overline{\mathfrak{S}}} \rho), (\sigma_2 \Box_{\overline{\mathfrak{S}}} \varepsilon) \qquad (9.63) $$

$$ \varepsilon \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi), (\sigma_3 \Box_{\overline{\mathfrak{S}}} \rho) \qquad (9.64) $$

$$ \rho \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi), (\sigma_4 \Box_{\overline{\mathfrak{S}}} \varepsilon). \qquad (9.65) $$

From $\pi \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_2 \Box_{\overline{\mathfrak{S}}} \rho)$ and $\pi \overline{\Box}_{\overline{\mathfrak{S}}} \sigma_2$, we deduce $(\sigma_2 \Box_{\overline{\mathfrak{S}}} \rho) = (\rho \Box_{\overline{\mathfrak{S}}} \pi)$.

From $\rho \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi)$ and $\sigma_4 \overline{\Box}_{\overline{\mathfrak{S}}} \rho$, we deduce $(\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi) = (\rho \Box_{\overline{\mathfrak{S}}} \pi)$.

From $\varepsilon \overline{\Box}_{\overline{\mathfrak{S}}} (\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi)$ and $\sigma_3 \overline{\Box}_{\overline{\mathfrak{S}}} \varepsilon$, we deduce $(\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi) = (\varepsilon \Box_{\overline{\mathfrak{S}}} \pi)$.

On another part, from (6.3) (if $\lambda \in (\mathfrak{S} \setminus \overline{\mathfrak{S}})$) or from (6.34) (if $\lambda \in \overline{\mathfrak{S}}^{\text{pure}}$), we deduce

$$ (\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi) \Box_{\overline{\mathfrak{S}}} (\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi) \overline{\Box}_{\overline{\mathfrak{S}}} \pi. \qquad (9.66) $$

As long as $\sigma_2 \neq \sigma_3$, we have $\sigma_3 \overline{\Box}_{\overline{\mathfrak{S}}} \pi_2$. Analogously, we have $\sigma_4 \overline{\Box}_{\overline{\mathfrak{S}}} \pi_2$. Hence,

$$ (\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi) = (\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi) = (\sigma_3 \Box_{\overline{\mathfrak{S}}} \pi) \Box_{\overline{\mathfrak{S}}} (\sigma_4 \Box_{\overline{\mathfrak{S}}} \pi) \overline{\Box}_{\overline{\mathfrak{S}}} \pi. \qquad (9.67) $$

As a result, we obtain

$$ (\varepsilon \Box_{\overline{\mathfrak{S}}} \pi) = (\rho \Box_{\overline{\mathfrak{S}}} \pi). \qquad (9.68) $$

Hence, we have obtained a contradiction with the property (6.4).

As a conclusion, this configuration is excluded.

As a conclusion of the previous analysis we obtain that $\forall i \in \{1, 2, 3, 4\}, \sigma_i \in \overline{\mathfrak{S}}^{\text{pure}}$.
Statements and Declarations

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Eric Buffenoir

Université de la Côte d'Azur, CNRS, InPhyNi, FRANCE

e-mail: eric.buffenoir@cnrs.fr