\[
  F \leq G \Longrightarrow F \ll G
\]

\[
  F = \mathop{\mathrm{Sup}}_{X \in \widetilde{F}} X \qquad \text{où} \qquad \widetilde{F} = \lbrace X \in \mathcal{M} \mid X \leq F \rbrace
\]

\[
  F \between G \overset{\mathrm{déf}}{\Longleftrightarrow} F \vee G \text{ existe, i.e.\ } \lbrace F, G \rbrace \text{ majoré dans } (\mathfrak{F}, \leq)
\]

\[
  \begin{cases}
    X \between X & \text{réflexive, symétrique} \\
    X \between Y,\ X' \leq X,\ Y' \leq Y \Longrightarrow X' \between Y'
  \end{cases}
\]

\[
  F \ll G \Longleftrightarrow \forall\, X \in \widetilde{F},\ \exists\, Y \in \widetilde{G},\ \text{t.q.\ } X \ll Y
\]

\[
  \widetilde{\mathfrak{F}}_{\mathrm{tf}} = \text{ensemble des parties fermées } \Phi \text{ de t.f.\ de } (\mathcal{M}, \leq) \text{ telles que } X, Y \in \Phi \Rightarrow X \between Y .
\]

\[
  \mathfrak{F}^{*} = \lbrace \mathrm{Multomb}(F) \mid F \in \mathfrak{F} \rbrace \subset \mathrm{Fig}(\mathcal{M})
\]

\[
  \mathrm{Multomb}(F) = \lbrace \mathrm{Omb}(X) \mid X \in \widetilde{F} \rbrace , \qquad
  \mathrm{Omb}\, X = \lbrace Y \in \mathcal{M} \mid Y \ll X \rbrace = \mathcal{M}_{\ll X} .
\]

\[
  F \parallel G \overset{\mathrm{déf}}{\Longleftrightarrow} F \between G \text{ et } F \cap G = \varnothing_{\mathfrak{F}}
\]

\[
  F \parallel G \Longleftrightarrow \forall\, X \in \widetilde{F},\ Y \in \widetilde{G},\ X \parallel Y
\]

\[
  X \underset{\mathcal{M}}{\parallel} Y \Longleftrightarrow X \underset{\mathcal{M}}{\between} Y \text{ et } \lbrace X, Y \rbrace \text{ n'est pas minoré dans } \mathcal{M}, \leq \text{, i.e.\ } \nexists\, Z \text{ avec } Z \leq X,\ Z \leq Y
\]

\[
  X' \ll X,\ X' \parallel K,\ Y' \ll Y,\ Y' \parallel K \Longrightarrow X' \parallel Y' .
\]

\[
  X \ll F,\ Y \ll G,\ X \parallel K,\ Y \parallel K \Longrightarrow X \parallel Y .
\]

\[
  F' \between G' \Longleftrightarrow F'_{L} \between G'_{L}
\]

\[
  F' \parallel G' \Longleftrightarrow F'_{L} \parallel G'_{L}
\]

\[
  F' \parallel G' \Longleftrightarrow F'_{L} \parallel G'_{L}
\]

\[
  L = F \cap G = \mathop{\mathrm{Sup}}_{Z \in \widetilde{F} \cap \widetilde{G}} Z .
\]

\[
  \mathrm{Multomb}(F) \overset{\mathrm{pol}}{\ll} \mathrm{Multomb}\, G \Longrightarrow F \ll G .
\]

\[
  \forall\, x, y, x', y' \in I \text{ tels que } x', y' \leq x, y ,\quad \exists\, z \in I \text{ tel que } x', y' \leq z \leq x, y .
\]

\[
  X'_{L} \between Y'_{L} \Longrightarrow X' \between Y' .
\]

\[
  F' \between G' \Longleftrightarrow F'_{L} \between G'_{L} .
\]

\[
  X'_{L} \parallel Y'_{L} \Longrightarrow X' \parallel Y'
\]

\[
  F' \parallel G' \Longleftrightarrow F'_{L} \parallel G'_{L}
\]

\[
  X' \parallel Y' .
\]

\[
  \begin{array}{l}
    \mathrm{At\ D} + \mathrm{At\ spéc} \Longrightarrow \\
    \mathrm{At\ D} + \mathrm{At\ pol} \Longrightarrow
  \end{array}
  \quad \mathrm{At\ C} \Longrightarrow \mathrm{At\ D} \Longrightarrow \mathrm{At\ D}_{0} .
\]

\[
  x \mathrel{\overline{\parallel}} Y \Longleftrightarrow x \ll Y \quad \text{i.e.\ } x \in \mathrm{omb}(Y) \qquad (\Longleftarrow \text{ est évident})
\]

\[
  x \parallel Y \Longleftrightarrow x \notin \mathrm{omb}(Y)
\]

\[
  F \parallel G \Longleftrightarrow \mathrm{Omb}\, F \cap \mathrm{Omb}\, G = \varnothing
\]

\[
  x \parallel F \Longleftrightarrow x \not\ll F \quad \text{i.e.\ } x \notin \mathrm{omb}(F)
\]

\[
  x \parallel y \Longleftrightarrow x \neq y .
\]

\[
  \mathrm{At\ ens\ D} \Longrightarrow \mathrm{At\ supp} .
\]

\[
  X \parallel Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X),\ y \in \mathrm{omb}\, Y,\ \text{on a } x \parallel y
\]

\[
  F \parallel G \Longleftrightarrow \forall\, x \in \mathrm{omb}(F),\ y \in \mathrm{omb}(G),\ \text{on a } x \parallel y
\]

\[
  x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ} \Longrightarrow x \parallel y
\]

\[
  (\mathrm{AL}) \qquad \forall\, x \in \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L),\ y \in \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L), \ \text{on a}\ x \parallel y
\]

\[
  (\mathrm{A}) \qquad \forall\, X' \in \mathrm{Omb}(X),\ Y' \in \mathrm{Omb}(Y),\ X'_L = Y'_L = \emptyset, \ \text{on a}\ X \parallel Y
\]

\[
  x_P = \emptyset \iff x \notin \mathrm{omb}(P) \ \text{i.e.}\ x \not\ll P
\]

\[
  x \ll P \iff x_P \neq \emptyset_{\mathfrak{F}} ,
\]

\[
  F_P = \emptyset \iff F \parallel P .
\]

\[
  \begin{array}{ll}
    a) & \text{At L 2} + \text{At CL (spéc) (ou pol)} \ \Longleftrightarrow\ \text{At L 2} + \text{At C (spéc) (ou pol)} \\
    b) & \text{At D}_0 + \text{At L 1} + \text{At CL (spéc) (ou pol)} \ \Longrightarrow\ \text{At L 2} \\
    c) & \text{At L 2} + \text{At L 3} \ \Longrightarrow\ \text{At D} \ \Longrightarrow\ \text{At D}_0 \\
    d) & \text{At L 3} + \text{At CL (spéc)} \ \Longrightarrow\ \text{At C} \\
    e) & \text{At D} \ \Longrightarrow\ \text{At L 3}
  \end{array}
\]

\[
  \begin{array}{lll}
    C \Rightarrow D \Rightarrow D_0 , & \text{At C (spéc)} & \text{At CL (spéc)} \\
    \quad \Downarrow & \text{At C (pol)} & \text{At CL (pol)} \\
    {[L 1]}\ L 2,\ L 3 & &
  \end{array}
\]

\[
  \text{At L 2} + \text{At L 3} + \text{At CL (spéc)} \quad
  \begin{array}{l}
    \text{implique tous les axiomes de compatibilité-} \\
    \text{disjonction (style spéc) :} \\
    \text{At C (spéc) \quad par (a)} \\
    \text{At D \quad par (c) d'où D}_0 \\
    \text{At C \quad par NB p.~17 ou par d)}
  \end{array}
\]

\[
  \begin{array}{l}
    \text{At L 1} + \text{At L 2} + \text{At L 3} + \text{At CL (spéc)} \\
    \quad \Updownarrow \\
    \text{At D}_0 + \text{At L 1} + \text{At L 3} + \text{At CL (spéc)} \\
    \quad \Downarrow \\
    \text{At D} + \text{At L 1} + \text{At CL (spéc)} \qquad \text{At C (spéc)}
  \end{array}
\]

\[
  (1) \qquad (\mathcal{M}, \leq, \ll)
\]

\[
  (2) \qquad \mathcal{L} = \lbrace x \in \mathcal{M} \mid x \ \text{minimal pour}\ \ll \rbrace ,
\]

\[
  (3) \qquad X \between Y \iff \forall\, x \in \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L),\ y \in \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L), \ \text{on a}\ x \parallel y ,
\]

\[
  L = X \cap Y = \operatorname*{Sup}_{Z \in \widetilde{X} \cap \widetilde{Y}} Z \qquad \Bigl(\text{donc}\ \mathrm{omb}(L) \overset{\mathrm{def}}{=} \bigcup_{Z \in \widetilde{X} \cap \widetilde{Y}} \mathrm{omb}(Z)\Bigr) .
\]

\[
  (4) \qquad X \between Y,\ X' \leq X,\ Y' \leq Y \Longrightarrow X' \between Y'
\]

\[
  (\mathrm{ML}\ 0) \qquad X \leq Y \Longrightarrow X \ll Y \qquad \text{pour mémoire}
\]

\[
  \begin{aligned}
    \mathrm{omb}(X) &= \lbrace x \in \mathcal{L} \mid x \ll X \rbrace \\
    \mathrm{omb}(X)^{\circ} &= \mathrm{omb}(X) \smallsetminus \bigcup_{Y < X} \mathrm{omb}(Y)
  \end{aligned}
\]

\[
  x \mathrel{\overset{\circ}{\ll}} X \overset{\mathrm{def}}{\iff} x \in \mathrm{omb}(X)^{\circ} \quad \text{i.e.} \quad x \ll X, \ \text{et}\ x \ll X' \leq X \Rightarrow X' = X .
\]

\[
  Y \mathrel{\overset{\circ}{\ll}} X \iff Y \ll X, \ \text{et}\ Y \ll X' \leq X \Rightarrow X' = X ,
\]

\[
  x \mathrel{\overset{\circ}{\ll}} Y \ \text{ssi}\ Z \mathrel{\overset{\circ}{\ll}} Y .
\]

\[
  x \mathrel{\overset{\circ}{\ll}} X' \iff Z \mathrel{\overset{\circ}{\ll}} X' .
\]

\[
  x \mathrel{\overset{\circ}{\ll}} Z \mathrel{\overset{\circ}{\ll}} X' \Longrightarrow x \mathrel{\overset{\circ}{\ll}} X' .
\]

\[
  \begin{array}{c}
    (v) \\ \Downarrow \\ (i) \Longleftrightarrow (iv)
  \end{array}
  \qquad (ii) \Longrightarrow (iii) \qquad \text{triviales.}
\]

\[
  Z \ll X'' \leq X \Longrightarrow X' \leq X'' .
\]

\[
  (1) \qquad X \between Y \overset{\mathrm{def}}{\iff} \forall\, x \in \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L),\ y \in \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L) \ \text{on a}\ x \parallel y
\]

\[
  \mathrm{omb}(L) \overset{\mathrm{déf}}{=} \bigcup_{\substack{Z \in \widetilde{X} \cap \widetilde{Y} \\ \text{i.e.}\ Z \leq X, Y}} \mathrm{omb}(Z)
\]

\[
  \mathrm{omb}(X) \cap \mathrm{omb}(Y) = \mathrm{omb}(L)
\]

\[
  \mathrm{omb}(X)^{\circ} \cap \mathrm{omb}(Y) \neq \emptyset \Longrightarrow X \leq Y .
\]

\[
  x \mathrel{\overset{\circ}{\ll}} X, \quad x \ll Y
\]

\[
  x \ll Z \leq X, Y
\]

\[
  \begin{cases}
    x \in \mathrm{omb}(X') \smallsetminus \mathrm{omb}(L') \\
    y \in \mathrm{omb}(Y') \smallsetminus \mathrm{omb}(L') \\
    \qquad \text{où}\ \mathrm{omb}(L') \overset{\mathrm{déf}}{=} \bigcup_{Z \in \widetilde{X}' \cap \widetilde{Y}'} \mathrm{omb}(Z)
  \end{cases}
\]

\[
  \exists\, Z \in \widetilde{X} \cap \widetilde{Y} \quad (\text{i.e.}\ Z \leq X, Y) \ \text{avec}\ x \ll Z
\]

\[
  x \ll T \leq S, Y'
\]

\[
  T \leq X', Y' \qquad \text{puisque}\ S \leq X'
\]

\[
  Z \mathrel{\overset{\circ}{\ll}} X, \quad Z \mathrel{\overset{\circ}{\ll}} Y, \quad X \between Y
\]

\[
  Z \mathrel{\overset{\circ}{\ll}} X, \ Z \ll Y \Longrightarrow X \leq Y
\]

\[
  x \mathrel{\overset{\circ}{\ll}} Z \mathrel{\overset{\circ}{\ll}} X
\]

\[
  x \between y \ \text{ssi}\ x = y \ \text{ou}\ x \parallel y
\]

\[
  \underbrace{X \between Y \ \text{et}\ \widetilde{X} \cap \widetilde{Y} = \emptyset}_{\text{i.e.}\ X \parallel Y \ \text{au sens de l'atelier}}
  \iff \forall\, x \in \mathrm{omb}(X),\ y \in \mathrm{omb}(Y) \Longrightarrow x \parallel y
\]

\[
  F \ll G \overset{\mathrm{déf}}{\iff} \forall\, X \in F,\ \exists\, Y \in G \ \text{avec}\ X \ll Y .
\]

\[
  \mathfrak{F} \subset \underline{\Phi}
\]

\[
  a)\ \mathfrak{F} = \mathfrak{F}_0 = \lbrace \text{réunions finies d'ensembles de la forme}\ \widetilde{X} \rbrace
\]

\[
  b)\ \text{toute partie fermée d'un}\ \widetilde{X}\ \text{\add{(pour $\leq$)} est de type fini}
\]

\[
  (\mathrm{ML}\ 6) \quad \text{Soit}\ X, Y \in \mathcal{M}.\ \text{Alors} \quad X \parallel Y \iff \forall\, x, y \in \mathcal{L},\ x \ll X,\ y \ll Y \Rightarrow x \parallel y .
\]

\[
  \Gamma_0 = \Bigl\lbrace \lbrace x, y \rbrace \in \mathfrak{P}_2(\mathcal{L}) \Bigm| \exists\, X, Y \in \mathcal{M},\ Y \leq X,\ y \ll Y,\ x \ll X,\ x \not\ll Y \Bigr\rbrace
\]

\[
  \Gamma_0 \subset \Gamma \subset \mathfrak{P}_2(\mathcal{L}) .
\]

\[
  \begin{aligned}
    \mathrm{omb}_{\Lambda}(X) &= \lbrace Z \in \Lambda \mid Z \ll X \rbrace \\
    \mathrm{omb}_{\Lambda}(X)^{\circ} &= \lbrace Z \in \Lambda \mid Z \mathrel{\overset{\circ}{\ll}} X \rbrace .
  \end{aligned}
\]

\[
  X \parallel Y \overset{\mathrm{déf}}{\iff} X \between Y, \ \text{et}\ \widetilde{X} \cap \widetilde{Y} = \emptyset
\]

\[
  \mathrm{omb}_{\Lambda}(X) \cap \mathrm{omb}_{\Lambda}(Y) = \mathrm{omb}_{\Lambda}(L) \overset{\mathrm{def}}{=} \bigcup_{Z \in \widetilde{X} \cap \widetilde{Y}} \mathrm{omb}_{\Lambda} Z
\]

\[
  \mathrm{omb}_{\Lambda}(Y) \cap \mathrm{omb}_{\Lambda}(X') = \bigcup_{Y' \in \struck{\widetilde{Y} \cap \mathrm{Omb}(X')}} \mathrm{omb}_{\Lambda}(Y')
  \qquad \overset{\text{si } Y \leq X}{=} \qquad \bigcup_{Y' \in \widetilde{Y} \cap \widetilde{X}'} \mathrm{omb}_{\Lambda}(Y')
\]

\[
  \mathrm{omb}_{\Lambda}(X) \cap \mathrm{omb}_{\Lambda}(Y) = \bigcup_{Z' \in \widetilde{X} \cap \widetilde{Y}} \mathrm{omb}_{\Lambda}(Z')
\]

\[
  \mathcal{R}(X, Y) \iff
  \begin{array}{l}
    \forall\, Z \in \mathcal{M} \text{ avec } Z \ll X,\ Z \ll Y, \\
    \exists\, Z' \in \mathcal{M} \text{ avec } Z \ll Z' \leq X, Y \\
    \text{i.e. } \mathrm{Omb}(X) \cap \mathrm{Omb}(Y) = \bigcup_{Z' \in \widetilde{X} \cap \widetilde{Y}} \mathrm{Omb}(Z')
  \end{array}
\]

\[
  X \between Y \Longrightarrow \mathcal{R}(X, Y)
\]

\[
  X \between Y \iff
  \begin{array}{l}
    \forall\, X' \in \mathrm{omb}_{\Lambda} X,\ Y' \in \mathrm{omb}_{\Lambda} Y \text{ avec } X'_L = Y'_L = \emptyset , \\
    \text{on a } X' \parallel Y'
  \end{array}
\]

\[
  X'_L \parallel Y'_L \Longrightarrow X' \parallel Y' .
\]

\[
  X \between Y \iff \forall\, X' \in \mathrm{omb}_{\Lambda}(X),\ Y' \in \mathrm{omb}_{\Lambda}(Y),\ X'_L = Y'_L = \emptyset \Longrightarrow X' \parallel Y'
\]

\[
  X \between Y \iff
  \begin{cases}
    \text{(i) Comme dans At C$\Lambda$ (spéc) ci-dessus} \\
    \text{(ii) $L = \emptyset$ ou $L \in \mathcal{M}$}
  \end{cases}
\]

\[
  \begin{array}{ccccc}
    \Lambda 3 + C\Lambda^{*} & \Longrightarrow & C\,\struck{\ill{}} & & \\
     & & \Downarrow & & \\
    \Lambda 2 + \Lambda 3 & \Longrightarrow & D & \Longrightarrow & \Lambda 3 \\
    \Downarrow & & \Downarrow & & \\
    \Lambda 2 & \Longrightarrow & D_0 & &
  \end{array}
\]

\[
  D_0 + \Lambda 1 + C\Lambda^{*} \Longrightarrow \Lambda 2 , \qquad
  \Lambda 2 + C\Lambda^{*} \iff \Lambda 2 + C^{*}
\]

\[
  \underset{\substack{\Downarrow \\ D_0}}{C,\ D},\ C^{*},\ \Lambda 1,\ \Lambda 2,\ \Lambda 3,\ C\Lambda^{*}
\]

\[
  \begin{array}{ll}
    a) & \Lambda 1,\ \Lambda 2,\ \Lambda 3,\ C\Lambda^{*} \\
    b) & \Lambda 1,\ \Lambda 2,\ D,\ C^{*} \\
    c) & \Lambda 1,\ C\Lambda^{*},\ D \\
    d) & \Lambda 1,\ \Lambda 2,\ C^{*},\ D_0
  \end{array}
\]

\[
  (\mathcal{M},\ \underset{\mathcal{M}}{\leq},\ \underset{\mathcal{M}}{\ll},\ \Lambda \subset \mathcal{M},\ \underset{\Lambda}{\parallel})
\]

\[
  \begin{cases}
    L = \widetilde{X} \cap \widetilde{Y} \\
    X'_L = \widetilde{X}' \cap \mathrm{Omb}(L) , \quad Y'_L = \widetilde{Y}' \cap \mathrm{Omb}(L) \\
    \text{où } \mathrm{Omb}(L) \overset{\mathrm{def}}{=} \bigcup_{Z \in L} \mathrm{Omb}(Z)
  \end{cases}
\]

\[
  \begin{array}{l}
    X \underset{\mathcal{M}}{\between} Y \iff \forall\, X' \in \mathrm{omb}_{\Lambda}(X),\ Y' \in \mathrm{omb}_{\Lambda}(Y) \quad (X'_L = \emptyset,\ Y'_L = \emptyset) \Longrightarrow X' \underset{\Lambda}{\parallel} Y' \\
    X \underset{\mathcal{M}}{\parallel} Y \iff \forall\, X' \in \mathrm{omb}_{\Lambda}(X),\ Y' \in \mathrm{omb}_{\Lambda}(Y) ,\ \text{on a } X' \underset{\Lambda}{\parallel} Y' .
  \end{array}
\]

\[
  \begin{aligned}
    P \parallel Q &\overset{\mathrm{def}}{\iff} \forall\, X \in P,\ \forall\, Y \in Q,\ \text{on a } X \parallel Y \\
    &\iff \forall\, X' \in \mathrm{omb}_{\Lambda} P,\ Y' \in \mathrm{omb}_{\Lambda}(Q),\ \text{on a } X' \parallel Y'
  \end{aligned}
\]

\[
  X'_L \parallel Y'_L \iff X' \parallel Y'
\]

\[
    \mathrm{omb}(Y) \cap \mathrm{omb}(X') = \mathrm{omb}(Y_{X'}) \overset{\mathrm{(def)}}{=} \bigcup_{Z \in Y_{X'}} \mathrm{omb}(Z) \quad \text{où } Y_{X'} = \widetilde{Y} \cap \mathrm{Omb}(X') .
  \]

\[
  X \between Y \iff
  \begin{cases}
    \forall\, x \in \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L),\ y \in \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L) \ (\text{où } L = \widetilde{X} \cap \widetilde{Y}) \text{ on a } x \parallel y \\
    [+\ L = \emptyset \text{ ou } L \text{ de la forme } \widetilde{Z}, \text{ dans le cas « polyédral »}]
  \end{cases}
\]

\[
  X \ll Y \iff X \leq Y , \quad \text{et } X \between Y \text{ tjrs satisfaits.}
\]

\[
  X \overset{\circ}{\ll} Y \iff X = Y
\]

\[
  \mathfrak{F} = \mathfrak{P}_{\mathrm{f}}(\mathcal{M}, \leq) \quad \text{ensemble de \emph{toutes} les parties fermées}
\]

\[
  \varphi : \mathfrak{F} \longrightarrow \mathrm{Fig}(\mathcal{L}) \qquad F \longmapsto \mathrm{multomb}(F)
\]

\[
  \mathrm{omb}(X) \cap \mathrm{omb}(Y) = \struck{\emptyset}\ \mathrm{omb}(L)
\]

\[
  |\varphi(X)| \cap |\varphi(Y)| = |\varphi(L)|
\]

\[
  \varphi_{\Lambda} : \mathfrak{F} \longrightarrow \mathrm{Fig}(\Lambda) \qquad F \longmapsto \mathrm{multomb}_{\Lambda}(F)
\]

\[
  \mathrm{Ssfig}(F) \xrightarrow{\ \sim\ } \mathrm{Ssfig}(\varphi_{\Lambda}(F)) , \quad F' \longmapsto \varphi_{\Lambda}(F')
\]

\[
  i : \Lambda \longrightarrow \mathcal{M}
\]

\[
  \mathrm{omb}(X) = \mathrm{omb}(Y) \quad (= |\varphi(X)| = |\varphi(Y)|)
\]

\[
  \mathcal{M} = \text{simplexes \emph{affines} dans } X
\]

\[
  |\varphi(F)| \cap |\varphi(G)| = |\varphi(L)| \quad \text{i.e.} \quad \mathrm{omb}(F) \cap \mathrm{omb}(G) = \mathrm{omb}(L) ,
\]

\[
    \mathrm{omb}(Y) \cap \mathrm{omb}(X') = \mathrm{omb}(Y') \Bigl( \overset{\mathrm{def}}{=} \bigcup_{Z \in Y_{X'}} \mathrm{omb}(Z) \Bigr) ,
  \]

\[
  (*) \qquad \Sigma_{\mathfrak{F}} \rightleftarrows \Sigma_{\mathcal{M}} , \qquad
  \mathfrak{S} \longmapsto \mathfrak{S} \cap \mathcal{M} , \qquad
  \mathfrak{F}_{S} = \lbrace F \in \mathfrak{F} \mid \widetilde{F} \subset S \rbrace \longleftarrow\!\shortmid\ S
\]

\[
  \operatorname{cosupp} A = \lbrace G \in \mathfrak{F} \mid \forall F \in A,\ F \parallel G \rbrace \subset \mathfrak{F}
\]

\[
  \operatorname{cosupp} A \in \Sigma_{\mathfrak{F}}
\]

\[
  \operatorname{supp} A = \operatorname{cosupp}(\operatorname{cosupp} A) ,
\]

\[
  A \longmapsto \operatorname{cosupp} A \qquad \mathfrak{P}(\mathfrak{F}) \to \Sigma^{\circ}_{\mathfrak{F}}
\]

\[
  \Sigma^{\circ}_{\mathfrak{F}} \longrightarrow \Sigma^{\circ}_{\mathfrak{F}}
\]

\[
  \complement A = \operatorname{cosupp} A ,
\]

\[
  \operatorname{Inf}^{\Sigma^{\circ}_{\mathfrak{F}}}_{i} A_i = \bigcap A_i
\]

\[
  \operatorname{Sup}^{\Sigma^{\circ}_{\mathfrak{F}}}_{i} A_i = \overline{\textstyle\bigcup A_i} \quad \Longrightarrow \ (= \operatorname{supp}(\textstyle\bigcup A_i))
\]

\[
  A \cap \complement A = \emptyset_{\Sigma}
\]

\[
  P \wedge \bigvee_i Q_i = \bigvee_i (P \wedge Q_i)
\]

\[
  \Sigma^{\circ}_{\mathcal{M}} \rightleftarrows \Sigma^{\circ}_{\mathfrak{F}} \qquad \text{iso.\ d'ens.\ ord.}
\]

\[
  F \parallel G \iff \forall X \in \mathrm{omb}_{\Lambda}(F),\ X \parallel G
\]

\[
  \mathrm{omb}_{\Lambda}(A) \overset{\mathrm{déf}}{=} \bigcup_{F \in A} \mathrm{omb}_{\Lambda}(F) \subset \Lambda ,
\]

\[
  \operatorname{cosupp}_{\mathfrak{F}} A = \operatorname{cosupp}_{\mathfrak{F}} \mathrm{omb}_{\Lambda} A .
\]

\[
  \mathrm{omb}_{\Lambda}(A) = \Lambda \cap A \qquad (A \text{ fermé dans } \mathfrak{F}, \ll)
\]

\[
  \operatorname{cosupp}_{\mathfrak{F}}(A) = \operatorname{cosupp}_{\mathfrak{F}}(A \cap \Lambda) \qquad \text{si } A \text{ fermé dans } \mathfrak{F} \text{ pour } \ll
\]

\[
  \operatorname{supp}_{\mathfrak{F}}(A) = \operatorname{supp}_{\mathfrak{F}}(A \cap \Lambda) \qquad \text{si } A \text{ fermé dans } \mathfrak{F} \text{ pour } \ll
\]

\[
  \operatorname{supp}_{\mathfrak{F}}(A) = \operatorname{supp}_{\mathfrak{F}}(\mathrm{omb}_{\Lambda}(A))
\]

\[
  \Sigma^{\circ}_{\mathfrak{F}} \rightleftarrows \Sigma^{\circ}_{\Lambda} , \qquad
  A \longmapsto A \cap \Lambda = \mathrm{omb}_{\Lambda}(A) , \qquad
  \operatorname{supp}_{\mathfrak{F}} S = \lbrace F \in \mathfrak{F} \mid \mathrm{omb}_{\Lambda}(F) \subset S \rbrace \longleftarrow\!\shortmid\ S
\]

\[
  \Sigma_{\mathfrak{F}} \simeq \Sigma_{\mathcal{M}} \simeq \Sigma_{\mathcal{L}}
\]

\[
  \Sigma_{\mathcal{L}} = \mathfrak{P}(\mathcal{L}) ,
\]

\[
  \operatorname{supp} A = \mathrm{omb}(A) \overset{\mathrm{déf}}{=} \bigcup_{F \in A} \mathrm{omb}(F)
\]

\[
  \operatorname{cosupp} A = \complement_{\mathcal{L}} \operatorname{supp} A \qquad \text{(complémentaire ordinaire)}
\]

\[
  F \parallel G \iff (\mathrm{omb}\, F \cap \mathrm{omb}\, G = \emptyset)
\]

\[
  \left\lbrace
  \begin{array}{l}
    A \subset B \iff A \cap \complement(B) = \emptyset_{\Sigma} \\
    A \cap \bigvee_i B_i = \bigvee_i A \cap B_i \quad \text{dans } \Sigma \ —
  \end{array}
  \right.
\]

\[
  P \subset \mathcal{L}
\]

\[
  \Sigma_{P} \simeq \mathfrak{P}(P) , \qquad \text{donc} \quad A \longmapsto \lbrace x \in P \mid \operatorname{supp} x \subset A \rbrace
\]

\[
  \Sigma_{\mathfrak{F}\ (\text{ou } \mathcal{M},\ \text{ou } \mathcal{L})} \simeq \mathfrak{P}(P)
\]

\[
  \operatorname{supp} A = \mathrm{omb}_{\Lambda} A = \bigcup_{F \in A} \mathrm{omb}_{\Lambda}(F) .
\]

\[
  \Sigma \simeq \mathfrak{P}(P)
\]

\[
  \mathcal{M} = \mathcal{M}_0 \amalg \mathcal{M}_1
\]

\[
  X \leq Y \iff \left\lbrace
  \begin{array}{l}
    X = Y \ \text{ou} \\
    X \in \mathcal{M}_0,\ Y \in \mathcal{M}_1,\ X \in \partial Y
  \end{array}
  \right.
\]

\[
  X \ll Y \iff X \subset Y
\]

\[
  X \between Y \iff \text{on est dans un des trois cas mutuellement exclusifs}
\]

\[
  X = [a,b] \qquad (a < b)
\]

\[
  a_1 < a_2 < \cdots < a_n
\]

\[
  \lbrace a_1 \rbrace ,\ [a_1, a_2] ,\ \lbrace a_2 \rbrace ,\ [a_2, a_3] ,\ \ldots ,\ \lbrace a_n \rbrace
\]

\[
  \varphi : \widetilde{F'} \longrightarrow \widetilde{F}
\]

\[
  \varphi^{*} : \mathfrak{P}(\widetilde{F}) \longrightarrow \mathfrak{P}(\widetilde{F'}) .
\]

\[
  \mathfrak{P}(P) \quad \text{disons, où } P \text{ est}
\]

\[
  \mathcal{M} \longrightarrow \mathfrak{P}(P) \qquad X \longmapsto |X|
\]

\[
  x \sim y \iff \mathcal{M}_x = \mathcal{M}_y \qquad (\text{où } \mathcal{M}_x = \lbrace X \in \mathcal{M} \mid x \in |X| \rbrace)
\]

\[
  \mathcal{L} \longrightarrow \mathfrak{P}(P) , \qquad x \longmapsto |x| ,
\]

\[
  \Sigma_{\mathcal{M}} \simeq \mathfrak{P}(P) \ ??
\]

\[
  |A| = \bigcup_{X \in A} |X|
\]

\[
  (*) \qquad \operatorname{supp} A \subset \operatorname{supp} B \overset{?}{\iff} |A| \subset |B|
\]

\[
  (\alpha) \qquad \operatorname{cosupp} B \subset \operatorname{cosupp} A ,
\]

\[
  \begin{array}{l}
  X \in \operatorname{cosupp} A \overset{\mathrm{déf}}{\iff} X \parallel A \overset{\mathrm{déf}}{\iff} \forall Y \in A,\ X \parallel Y \\
  \qquad \overset{\mathrm{iii}_{\mathcal{M}}}{\iff} \forall Y \in A,\ |X| \cap |Y| = \emptyset \iff |X| \cap |A| = \emptyset
  \end{array}
\]

\[
  (\cdot) \qquad \forall X \in \mathcal{M}, \quad |X| \cap |B| = \emptyset \Longrightarrow |X| \cap |A| = \emptyset .
\]

\[
  B = \operatorname{supp} B ,
\]

\[
  A \subset B
\]

\[
  \operatorname{supp} A \subset \operatorname{supp} B \Longleftarrow |A| \subset |B|
\]

\[
  (\Leftrightarrow A \subset \operatorname{supp} B)
\]

\[
  \Sigma_{\mathcal{M}} \longrightarrow \mathfrak{P}(P) \qquad A \longmapsto |A|
\]

\[
  \left| \bigwedge_i A_i \right| \subset \bigcap_i |A_i| \qquad \text{tautologiques}
\]

\[
  \left| \bigvee_i A_i \right| \supset \bigcup_i |A_i| \qquad \text{id}
\]

\[
  |\complement_{\mathcal{M}}(A)| \subset P \smallsetminus |A|
\]

\[
  \complement_{\mathcal{M}}(A) = \lbrace X \in \mathcal{M} \mid X \parallel A \rbrace = \lbrace X \in \mathcal{M} \mid |X| \cap |A| = \emptyset \rbrace
\]

\[
  |\complement_{\mathcal{M}}(A)| = \bigcup \overline{X} \quad \text{pour } X \in \mathcal{M} \text{ t.q.\ } |X| \subset P \smallsetminus A
\]

\[
  (**) \qquad |A| \overset{?}{=} |\operatorname{supp} A| ,
\]

\[
  |A| \subset |\operatorname{supp} A|
\]

\[
  |\operatorname{supp} A| \subset |A|
\]

\[
  \bigcup_{\substack{X \in \mathcal{M} \\ X \parallel A}} |X| = P \smallsetminus |A|
  \qquad (\text{i.e.\ } X \in \operatorname{cosupp} A)
\]

\[
  \forall A \subset \mathcal{M}, \qquad |\operatorname{cosupp} A| = P \smallsetminus |A| .
\]

\[
  |A| = |\operatorname{supp} A| ,
\]

\[
  \operatorname{supp} A \subset \operatorname{supp} B \iff |A| \subset |B|
\]

\[
  (\ \iff A \subset \operatorname{supp} B \iff \operatorname{cosupp} B \subset \operatorname{cosupp} A
\]

\[
  \Sigma_{\mathcal{M}} \longrightarrow \mathfrak{P}(P) \qquad A \longmapsto |A|
\]

\[
  (\times) \qquad \struck{\ill{}}\ \left| \bigcap A_i \right| \overset{?}{=} \bigcap_i |A_i|
\]

\[
  |\operatorname{cosupp} A| = P \smallsetminus |A| \qquad \forall A \subset \mathcal{M} ,
\]

\[
  X \longmapsto |X| : \mathcal{M} \longrightarrow \mathfrak{P}(P)
\]

\[
  |A| \overset{\mathrm{déf}}{=} \bigcup_{X \in A} |X| .
\]

\[
  |A| = \operatorname{supp} A .
\]

\[
  \Sigma_{\mathcal{M}} \rightleftarrows \mathfrak{P}(P) \qquad A \longmapsto |A|
\]

\[
  I^{\circ} = \left]0, 1\right[ \times \left]0, 1\right[ .
\]

\[
  z = (x, x') \in J^{\circ} = I \times I'
\]

\[
  A_z = \lbrace z \rbrace
\]

\[
  [z'', 1] \ \text{pour}\ z'' > z , \qquad [0, z'] \ \text{pour}\ z' < z .
\]

\[
  |z| = \lbrace x, x' \rbrace , \qquad |0| = \lbrace 0 \rbrace , \qquad |1| = \lbrace 1 \rbrace .
\]

\[
  B_z = \lbrace z' \in J \mid z' < z \ \text{ou}\ z' > z \rbrace
\]

\[
  |B_z| = J \smallsetminus |z|
\]

\[
  |B_{z_1} \cap B_{z_2}| \subsetneqq |B_{z_1}| \cap |B_{z_2}| .
\]

\[
  [x_1, x_2] \cup [x'_2, x'_1] = |A_{z_1} \vee A_{z_2}| \supsetneqq |A_{z_1}| \cup |A_{z_2}| = \lbrace x_1, x_2, x'_1, x'_2 \rbrace
\]

\[
  A = \lbrace z_1, z_2 \rbrace
\]

\[
  |A| = |z_1| \cup |z_2| = \lbrace x_1, x_2, x'_1, x'_2 \rbrace
\]

\[
  |\mathrm{supp}\, A| = [x_1, x_2] \cup [x'_1, x'_2]
\]

\[
  |A| \subsetneqq |\mathrm{supp}\, A| .
\]

\[
  A_{z_3} \cap (A_{z_1} \vee A_{z_2}) = A_{z_3}
\]

\[
  (\underbrace{A_{z_3} \cap A_{z_1}}_{\emptyset}) \vee (\underbrace{A_{z_3} \cap A_{z_2}}_{\emptyset}) = \emptyset
\]

\[
  X^{\circ} = \mathrm{supp}_{\mathcal{M}} X \cap \mathrm{cosupp}_{\mathcal{M}} \partial X \qquad (\subset \mathcal{M})
\]

\[
  \partial X = \lbrace Y \in \mathcal{M} \mid Y < X \rbrace = \widetilde{X} \smallsetminus \lbrace X \rbrace
\]

\[
  \mathrm{supp}\, X = \mathop{\mathrm{Sup}_{\Sigma}}_{Y \in \widetilde{X}} Y^{\circ} \qquad \text{Sup dans}\ \Sigma_{\mathcal{M}}
\]

\[
  \mathrm{cosupp}\, X = \bigcap_{Y \in \widetilde{X}} \complement_{\mathcal{M}}(Y^{\circ})
\]

\[
  Z \parallel X \Longleftrightarrow \forall\, Y \in \widetilde{X},\ Z \parallel Y^{\circ}
\]

\[
  (\delta) \qquad \mathrm{supp}\, \mathrm{omb}(X)^{\circ} \subset X^{\circ}
\]

\[
  Y \in X^{\circ} \overset{\mathrm{def}}{=} \mathrm{supp}\, X \cap \mathrm{cosupp}\, \partial X ,
\]

\[
  X^{\circ} = \lbrace Y \in \mathrm{Omb}(X) \mid Y_{\partial X} = \emptyset ,\ \text{i.e.}\ \widetilde{Y} \cap \mathrm{Omb}(\partial X) = \emptyset \rbrace
\]

\[
  Y_{\partial X} = \emptyset \Longleftrightarrow Y \parallel \partial X
\]

\[
    Z|L = \emptyset \Longrightarrow Z \parallel \partial X
  \]

\[
  Y \smallsetminus L \parallel Z
\]

\[
  Y \smallsetminus L \parallel Z \smallsetminus K
\]

\[
  X \parallel Y \Longleftrightarrow \forall\, X' \in \widetilde{X},\ Y' \in \widetilde{Y},\ X'^{\circ} \parallel Y'^{\circ}
\]

\[
  X \parallel Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X),\ \forall\, y \in \mathrm{omb}(Y),\quad x \parallel y
\]

\[
  X^{\circ} = \Big\lbrace Y \in \mathrm{Omb}(X) \ \Big|\ Y \parallel \partial X \ \text{i.e.}\ Y_{\partial X} = \emptyset \ \text{i.e.}\ \widetilde{Y} \cap \mathrm{Omb}(\partial X) = \emptyset \ \text{i.e.}\ \nexists\, Y' \leq Y,\ Y' \ll X' < X \Big\rbrace
\]

\[
  X^{\circ} \parallel Y^{\circ}
\]

\[
  \forall\, X' \in X^{\circ},\ Y' \in Y^{\circ},\ \text{on a}\ X' \parallel Y' .
\]

\[
  X \parallel Y \Longleftrightarrow \forall\, X' \in \widetilde{X},\ Y' \in \widetilde{Y},\ \text{on a}\ X'^{\circ} \parallel Y'^{\circ} .
\]

\[
  Y'|Y_{\partial X} \leq Y'|\partial Y
\]

\[
  \boxed{Y^{\circ} \neq \emptyset}
\]

\[
  Y \in X^{\circ} \Longrightarrow \mathrm{Omb}(Y) \subset X^{\circ}
\]

\[
  \bigcup_{Y' \in \widetilde{Y}} Y'^{\circ} \subset X^{\circ} , \quad \text{prouvons}\ Y \in X^{\circ} ,
\]

\[
  X^{\circ} \cap Y^{\circ} = \emptyset
\]

\[
  S_A = \mathrm{Supp}_{\mathcal{M}} \Big( \bigcup_{X \in A} X^{\circ} \Big) \in \Sigma_{\mathcal{M}}
  = \mathop{\mathrm{Sup}_{\Sigma_{\mathcal{M}}}}_{X \in A} \mathrm{Supp}(X^{\circ}) .
\]

\[
  S_A = \mathop{\mathrm{Sup}_{\Sigma_{\mathcal{M}}}}_{i} S_{A_i} .
\]

\[
  \mathfrak{P}(\widetilde{F}) \longrightarrow \Sigma_{\mathcal{M}} , \qquad A \longmapsto S_A
\]

\[
  S_{A \smallsetminus B} = S_A \cap \complement_{\mathcal{M}}(S_B)
\]

\[
  S_A = \mathrm{supp}\, G \overset{\mathrm{def}}{=} \mathrm{Supp}(\hat{A}) .
\]

\[
  S_{\widetilde{F}} \overset{\mathrm{def}}{=} \mathrm{Supp}\Big( \bigcup_{X \in \widetilde{F}} X^{\circ} \Big) = \mathrm{Supp}\, F
\]

\[
  Y \parallel F \Longleftrightarrow \forall\, X \in \widetilde{F},\ Y \parallel X^{\circ}
\]

\[
  \big( Y \parallel \widetilde{F} \big)
\]

\[
  S_A \subset S_B \Longrightarrow A \subset B
\]

\[
  S_{\lbrace X \rbrace} \subset S_B \Longrightarrow X \in B
\]

\[
  \big( S_{\lbrace X \rbrace} = \mathrm{supp}(X^{\circ}) \big)
\]

\[
  X^{\circ} \subset S_B \Longrightarrow X \in B
\]

\[
  X \notin B \Longrightarrow \exists\, Z \in \mathcal{M},\ Z \parallel \bigcup_{Y \in B} Y^{\circ} \ \text{et}\ Z \overline{\parallel}\, X^{\circ}
\]

\[
  Z \in X^{\circ}
\]

\[
  X'_L \leq X'_{\partial X} = \emptyset , \qquad Y'_L \leq Y'_{\partial X} = \emptyset
\]

\[
  S \cap \complement S = \emptyset \ \ (\text{plus petit élt.\ de}\ \Sigma)
\]

\[
  S \vee \complement S = \mathcal{M} \qquad (\text{plus grand élt de}\ \Sigma) .
\]

\[
  S_J = \mathop{\mathrm{Sup}_{\Sigma}}_{i \in J} \mathrm{supp}\, A_i
\]

\[
  \varphi : J \longmapsto S_J : \mathfrak{P}(I) \longrightarrow \Sigma
\]

\[
  S_{A \smallsetminus B} = S_A \cap \complement S_B .
\]

\[
  S_A \subset S_B \Longrightarrow A \subset B
\]

\[
  S_{\lbrace i \rbrace} \subset S_B \Longrightarrow i \in B
\]

\[
  S_B \parallel S_C \qquad \text{a fortiori}\ S_B \cap S_C = \emptyset
\]

\[
  A = B \cup B' \qquad \text{où}\ B' = A \smallsetminus B
\]

\[
  S_A = S_B \vee S_{B'}
\]

\[
  S_{B'} = (S_B \vee S_{B'}) \cap \complement S_B ,
\]

\[
  S' = (S \vee S') \cap \complement S \qquad \big( = \lbrace X \in S \vee S' \mid X \parallel S \rbrace \big)
\]

\[
  \mathrm{Cosupp}\, A = \lbrace A', B \rbrace , \qquad \mathrm{supp}\, A = \mathrm{cosupp}\, \lbrace A', B \rbrace = A'
\]

\[
  \mathrm{Cosupp}\, B = \lbrace A, B' \rbrace , \qquad \mathrm{supp}\, B = B
\]

\[
  \text{Comme}\quad A \cup B = \complement A \cap \complement B = \emptyset \qquad \text{pas de sommet lié à $A$ et $B$ à la fois !}
\]

\[
  \mathrm{Supp}\, A \cup B = A \vee B = \mathcal{M} .
\]

\[
  S_{A \smallsetminus B} = S_A \cap \complement S_B
\]

\[
  X^{\circ} = \mathrm{Supp}\, \mathrm{omb}(X)^{\circ}
\]

\[
  S = \mathrm{supp}(S \cap \mathcal{L}) . \qquad (\text{plus gén., si $S$ est fermée pour $\ll$, $\mathrm{supp}\, S = \mathrm{Supp}(S \cap \mathcal{L})$}) .
\]

\[
  \varphi : \widetilde{F} \longrightarrow \widetilde{X} , \quad \text{soit} \quad \widetilde{F}^{\circ} = \varphi^{-1}(\lbrace X \rbrace) = \lbrace X' \in \widetilde{F} \mid X' \overset{\circ}{\ll} X \rbrace .
\]

\[
  X^{\circ} = \mathop{\mathrm{Sup}_{\Sigma}}_{X' \in \widetilde{F}^{\circ}} X'^{\circ}
  \quad \Big( = \mathrm{Supp} \bigcup_{X' \in \widetilde{F}^{\circ}} X'^{\circ} = \mathrm{Supp} \bigcup_{X' \in \widetilde{F}^{\circ}} \mathrm{omb}(X')^{\circ} \Big)
\]

\[
  F \ll X \quad \text{et} \quad \mathrm{supp}\, F = \mathrm{supp}\, X .
\]

\[
  |F| = |X| .
\]

\[
  |X^{\circ}| \subset |X| \smallsetminus |\partial X|
\]

\[
  X \mathrel{|{\circ}|} Y \Longleftrightarrow X^{\circ} \parallel Y^{\circ}
\]

\[
  \Sigma_{\mathcal{M}} \simeq \mathfrak{P}(S)
\]

\[
  X^{\circ} =: \mathrm{supp}\, X \cap \mathrm{cosupp}\, \partial X
\]

\[
  \leq, \quad \ll, \quad \mathrel{|{\circ}|}
\]

\[
  X \between Y \Longleftrightarrow \forall\, X' \in \widetilde{X} \smallsetminus L,\ Y' \in \widetilde{Y} \smallsetminus L,\quad X' \mathrel{|{\circ}|} Y'
\]

\[
  Z \ll Z' \leq X, Y .
\]

\[
  X \parallel Y \Longleftrightarrow \forall\, X' \in \widetilde{X},\ Y' \in \widetilde{Y}, \text{ on a } X' \mathrel{|{\circ}|} Y'
\]

\[
  X'_L = Y'_L = \emptyset
\]

\[
  X' \mathrel{\overset{\circ}{\ll}} X_1 \leq X, \qquad Y' \mathrel{\overset{\circ}{\ll}} Y_1 \leq Y
\]

\[
  X \mathrel{|{\circ}|} Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ}, \text{ on a } x \parallel y .
\]

\[
  X \mathrel{|{\circ}|} Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ}, \text{ on a } x \parallel y,
\]

\[
  X \between Y \Longleftrightarrow \underbrace{\forall\, x \in \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L),\ y \in \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L), \text{ on a } x \parallel y}_{X \between' Y}
\]

\[
  \mathrm{omb}(X) \smallsetminus \mathrm{omb}(L) = \bigcup_{X' \in \widetilde{X} \smallsetminus L} \mathrm{omb}(X')^{\circ}, \qquad
  \mathrm{omb}(Y) \smallsetminus \mathrm{omb}(L) = \bigcup_{Y' \in \widetilde{Y} \smallsetminus L} \mathrm{omb}(Y')^{\circ}
\]

\[
  \forall\, X' \in \widetilde{X} \smallsetminus L,\ Y' \in \widetilde{Y} \smallsetminus L, \text{ on a } \underbrace{\bigl[\forall\, x \in \mathrm{omb}(X')^{\circ},\ y \in \mathrm{omb}(Y')^{\circ} \text{ on a } x \parallel y\bigr]}_{X' \mathrel{|{\circ}|} Y'}
\]

\[
  (*) \qquad X \mathrel{|{\circ}|} Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ}, \text{ on a } x \parallel y
\]

\[
  (**) \qquad X \parallel Y \Longleftrightarrow \forall\, X' \in \widetilde{X},\ Y' \in \widetilde{Y}, \text{ on a } X' \mathrel{|{\circ}|} Y' .
\]

\[
  (***) \qquad X \mathrel{|{\circ}|} Y \Longleftrightarrow \forall\, x \in \mathrm{omb}(X)^{\circ},\ y \in \mathrm{omb}(Y)^{\circ}, \text{ on a } x \mathrel{|{\circ}|} y
\]

\[
  \mathrm{omb}(X)^{\circ} \neq \emptyset \qquad \forall\, X \in \mathcal{M},
\]

\[
  \leq, \quad \ll, \quad \mathrel{|{\circ}|}
\]

\[
  X \leq Y \Longrightarrow X \ll Y
\]

\[
  \widetilde{X}^{Z} = \lbrace X' \in \widetilde{X} \mid Z \ll X' \rbrace
\]

\[
  X \mathrel{|{\circ}|} Y,\ X' \mathrel{\overset{\circ}{\ll}} X,\ Y' \mathrel{\overset{\circ}{\ll}} Y \Longrightarrow X' \mathrel{|{\circ}|} Y'
\]

\[
  Y \neq Z \Longrightarrow Y \mathrel{|{\circ}|} Z
\]

\[
  X \between Y \overset{\mathrm{def}}{\Longleftrightarrow} \forall\, X' \in \widetilde{X} \smallsetminus \widetilde{X} \cap \widetilde{Y},\ Y' \in \widetilde{Y} \smallsetminus \widetilde{X} \cap \widetilde{Y}, \text{ on a } X' \mathrel{|{\circ}|} Y'
\]

\[
  X \parallel Y \overset{\mathrm{def}}{\Longleftrightarrow} \forall\, X' \in \widetilde{X},\ Y' \in \widetilde{Y}, \text{ on a } X' \mathrel{|{\circ}|} Y'
\]

\[
  X \parallel Y \Longleftrightarrow X \between Y \text{ et } \widetilde{X} \cap \widetilde{Y} = \emptyset \quad (\text{i.e.\ } \lbrace X, Y \rbrace \text{ non minoré dans } \mathcal{M})
\]

\[
  \widetilde{F} = \lbrace X \in \mathfrak{F} \mid X \leq F,\ X \text{ « strictement irréductible »} \rbrace
\]

\[
  \forall\, X \in \widetilde{F} \smallsetminus \widetilde{F} \cap \widetilde{G},\ Y \in \widetilde{G} \smallsetminus \widetilde{F} \cap \widetilde{G} \Longrightarrow X \mathrel{|{\circ}|} Y
\]

\[
  \mathfrak{F} = \lbrace \Phi \in \mathfrak{P}_f(\mathcal{M}) \mid X, Y \in \Phi \Rightarrow X \between Y \rbrace ,
\]