\[
  A \wedge \operatorname{Sup}_i B_i = \operatorname{Sup}_i A \wedge B_i ,
\]

\[
  X \neq x \overset{?}{\Longrightarrow} \operatorname{Supp}^{\circ}(X) \not\subset \operatorname{Supp}^{\circ} x ,
  \quad \text{i.e.} \quad \operatorname{cosupp}^{\circ} x \not\subset \operatorname{cosupp}^{\circ} X ,
\]

\[
  \exists\, Z \in \mathcal{A} \quad \text{avec} \quad Z \mathrel{|o|} x ,\ Z \mathrel{\overline{|o|}} y .
\]

\[
  \mathfrak{M} = \mathfrak{M}_0 \amalg \mathfrak{M}_1 ,
\]

\[
  \text{(a)} \quad
  \begin{cases}
    \partial I_{\{a,b\}} = \{a,b\} \subset \mathfrak{M}_0 \\
    \partial x = \emptyset \subset \mathfrak{M}
  \end{cases}
  \qquad
  \text{(b)} \quad
  \begin{cases}
    \delta I_{a,b} = \partial I_{a,b} = \{a,b\} \\
    \delta x = \{x\}
  \end{cases}
\]

\[
  \partial X = \{ Y \in \mathfrak{M} \mid Y \lhd X \} \qquad (X \in \mathfrak{M}) .
\]

\[
  X \mathrel{\mathring{\ll}} Y \iff |X| \subset |Y| \ \text{et} \ X \mathrel{\overline{\lhd}} Y .
\]

\[
  X \mathrel{\mathring{\ll}} Y \iff
  \begin{cases}
    X = Y \\
    \text{ou } X = x \in \mathcal{L},\ Y = I_{a,b},\ a < x < b \\
    \text{ou } X = I_{a,b},\ Y = I_{a',b'},\ a' \leq a < b \leq b'
  \end{cases}
\]

\[
  X \ll Y \iff X \lhd Y \ \text{ou} \ X \mathrel{\mathring{\ll}} Y \iff |X| \subset |Y|
\]

\[
  \iff
  \begin{cases}
    X = Y \\
    \text{ou } X = x \in \mathcal{L},\ Y = I_{a,b},\ a \leq x \leq b \\
    \text{ou } X = I_{a,b},\ Y = I_{a',b'},\ a' \leq a < b \leq b'
  \end{cases}
\]

\[
  X < Y \overset{\mathrm{def}}{\iff}
  \begin{cases}
    X = x,\ Y = y,\ x, y \in \mathcal{L},\ x < y \\
    X = x,\ Y = I_{a,b},\ x \leq a < b \\
    X = I_{a,b},\ Y = y,\ a < b \leq y \\
    X = I_{a,b},\ Y = I_{a',b'},\ a < b \leq a' < b'
  \end{cases}
\]

\[
  X < Y \iff \bigl( X \neq Y \ \text{et} \ (\forall\, x \in \delta X,\ y \in \delta Y,\ \text{on a } x \leq y) \bigr)
  \iff \bigl( |X| \leq |Y| \bigr)
\]

\[
  \operatorname{or} X = \operatorname{or} \delta X , \qquad \operatorname{ex} X = \operatorname{ex} \delta X .
\]

\[
  X < Y \Longrightarrow X' \leq Y' , \ \struck{et}
\]

\[
  X < Y \Longrightarrow \bigl( X' < Y' \ \text{ou} \ X' = \operatorname{ex} X = Y' = \operatorname{or} Y \bigr) .
\]

\[
  X \mathrel{|o|} Y \overset{\mathrm{def}}{\iff} X < Y \ \text{ou} \ Y < X \quad \text{i.e.} \quad \{X, Y\} \in \operatorname{Drap}_2(\mathfrak{M}, \leq) .
\]

\[
  X \neq Y \Longrightarrow \widetilde{X} \cap \widetilde{Y} = \delta X \cap \delta Y , \ \text{de cardinal } 0 \text{ ou } 1
\]

\[
  X \asymp Y \iff X \overset{\sim}{=} Y \ \text{ou} \ X \mathrel{|o|} Y \quad \text{i.e.} \quad \{X, Y\} \in \operatorname{Drap}(\mathfrak{M})
\]

\[
  \widetilde{X} \cap \widetilde{Y} = \delta X \cap \delta Y \ \text{est de cardinal } 0 \text{ ou } 1 .
\]

\[
  \operatorname{Préfig}(\mathfrak{M}) = \operatorname{Drap}^{*}(\mathfrak{M}, \leq)
\]

\[
  \operatorname{Fig}(\mathfrak{M}) = \operatorname{Drap}^{*}(\mathfrak{M}, \leq) \cap \mathfrak{P}_f(\mathfrak{M}, \trianglelefteq)
\]

\[
  \delta \mathfrak{F}' \subset \delta \mathfrak{F} \cup \{\omega_0, \omega_1\}
\]

\[
  \widehat{\mathfrak{M}}_0 = \mathfrak{M}_0 = \mathcal{L} , \qquad
  \widehat{\mathfrak{M}}_1 = \mathfrak{M}_1 \amalg \mathfrak{M}_1^{-} \amalg \mathfrak{M}_1^{+}
\]

\[
  \mathfrak{M}_1^{-} = \{ I_{<a} \mid a \in \mathcal{L},\ a \text{ non minimal dans } \mathcal{L} \}
\]

\[
  \mathfrak{M}_1^{+} = \{ I_{>a} \mid a \in \mathcal{L},\ a \text{ non maximal dans } \mathcal{L} \}
\]

\[
  \partial I_{<a} = \delta I_{<a} = \text{\struck{$\ldots$}} = \{a\}
\]

\[
  |I_{<a}| \overset{\mathrm{def}}{=} \mathcal{L}_{\leq a} = \{ x \in \mathcal{L} \mid x \leq a \}
\]

\[
  |I_{<a}|^{\circ} \overset{\mathrm{def}}{=} |I_{<a}| \smallsetminus \partial I_{<a} = \mathcal{L}_{<a} = \{ x \in \mathcal{L} \mid x < a \}
\]

\[
  \begin{cases}
    I_{<a} \mathrel{\mathring{\ll}} I_{<b} \iff a \leq b \\
    X \mathrel{\mathring{\ll}} I_{<a} \ \text{si} \iff X \leq a \quad \text{(cf.\ déf.\ldots)}
  \end{cases}
\]

\[
  I_{<a} \mathrel{\mathring{\ll}} X \qquad \text{pour } X \in \mathfrak{M} .
\]

\[
  X < Y \iff |X| \leq |Y| \qquad \text{On trouve}
\]

\[
  \text{\struck{$X \mathrel{|o|} Y \iff X < Y$ ou $Y < X$, i.e.\ $\{X,Y\} \in \operatorname{Drap}_2(\widehat{\mathfrak{M}}, \leq)$}}
\]

\[
  X \leq Y \iff
  \begin{cases}
    1^{\circ})\ \delta X \leq \delta Y \\
    2^{\circ})\ \text{si $X$ ou $Y$ est de la forme $I_{<a}$, $I_{>b}$, on exige \struck{pour} ceci :} \\
    \qquad \text{a) si } Y = I_{<a},\ \text{alors } X = Y \\
    \qquad \text{b) si } X = I_{>a},\ \text{alors } Y = X
  \end{cases}
\]

\[
  X \leq Y \Longrightarrow |X| \leq |Y| , \qquad X < Y \Longrightarrow |X| < |Y| .
\]

\[
  I_{<a} \leq I_{>b} \iff a \leq b .
\]

\[
  X \mathrel{|o|} Y \iff X < Y \ \text{ou} \ Y < X \quad \text{i.e.} \quad \{X, Y\} \in \operatorname{Drap}_2(\widehat{\mathfrak{M}}, \leq) .
\]

\[
  X < Y,\ X' \mathrel{\mathring{\ll}} X,\ Y' \mathrel{\mathring{\ll}} Y \Longrightarrow X' < Y' .
\]

\[
  \struck{\ill{}} \quad X < Y,\ X' \ll X,\ Y' \ll Y \Longrightarrow X' \leq Y'
\]

\[
  \hookrightarrow X' < Y' \ \text{ou} \ X' = \operatorname{ex} X = Y' = \operatorname{or} Y .
\]

\[
  X = I_{<a} ,
\]

\[
  T = \delta \mathfrak{F} \in \operatorname{Drap}^{*}(\mathcal{L}) ,
\]

\[
  T = \{ t_1 < t_2 < \cdots < t_n \}
\]

\[
  [I_{<t_1}],\ t_1,\ I_{t_1,t_2},\ t_2,\ I_{t_2,t_3},\ \ldots,\ t_n,\ [I_{>t_n}]
\]

\[
  \forall\, t \in T, \quad t \in \mathfrak{F}, \ \text{ou} \ t \lhd X \ \text{avec} \ X \in \mathfrak{F} .
\]

\[
  \mathfrak{F}^{*} = \widehat{\mathfrak{F}} \smallsetminus \mathfrak{F}
\]

\[
  \delta \mathfrak{F}^{*} = \delta \mathfrak{F} \smallsetminus \mathfrak{F}_{\mathrm{red}} , \quad \text{soit } T'
\]

\[
  \mathfrak{F}_{\mathrm{red}} = \{ s \in \mathfrak{F}_0 \mid \operatorname{ord}(s, \mathfrak{F}) = 2 \} .
\]

\[
  (\mathfrak{F}^{**})_0 = \mathfrak{F}_0 \smallsetminus \mathfrak{F}_{\mathrm{red}}
\]

\[
  X_1,\ x_1,\ X_2,\ x_2,\ \ldots,\ x_{k-1},\ X_k
\]

\[
  X_1 < x_1 < X_2 < x_2 < \cdots < x_{k-1} < X_k
\]

\[
  \text{et} \quad x_1 \lhd X_1, X_2 ,\quad x_2 \lhd X_2, X_3 ,\ \ldots,\ x_{k-1} \lhd X_{k-1}, X_k .
\]

\[
  \operatorname{Omb}^{\circ}_{\mathfrak{M}}(\mathfrak{F}) = \bigcup_{X \in \mathfrak{F}} \operatorname{Omb}^{\circ}_{\mathfrak{M}}(X)
  = \{ Z \in \mathfrak{M} \mid \exists\, X \in \mathfrak{F},\ Z \mathrel{\mathring{\ll}} X \} .
\]

\[
  \operatorname{cosupp}^{\circ}_{\mathfrak{M}}(\mathfrak{F})
  = \operatorname{cosupp}^{\circ}\bigl(\operatorname{Omb}^{\circ}_{\mathfrak{M}}(\mathfrak{F})\bigr)
  = \operatorname{Omb}^{\circ}_{\mathfrak{M}}(\mathfrak{F}^{*}) .
\]

\[
  \operatorname{supp}^{\circ}_{\mathfrak{M}}(\mathfrak{F})
  = \operatorname{supp}^{\circ}_{\mathfrak{M}}\bigl(\operatorname{Omb}^{\circ}_{\mathfrak{M}}(\mathfrak{F})\bigr)
  = \operatorname{Omb}^{\circ}_{\mathfrak{M}}(\mathfrak{F}^{**}) .
\]

\[
  \{ I, \omega_0, \omega_1 \} \ \text{ou} \ (I, \partial I) , \quad \partial I = \{\omega_0, \omega_1\}
\]

\[
  \omega_0 < \omega_1 \quad \text{i.e.} \quad \operatorname{card} I \geq 2 .
\]

\[
  |I|^{\circ} = \emptyset , \quad \text{où} \quad |I|^{\circ} = I \smallsetminus \delta I \subset I .
\]

\[
  \mathfrak{F}^{*}_{\mathrm{is}} = \mathfrak{F}_{\mathrm{lac}} , \qquad \mathfrak{F}^{*}_{\mathrm{lac}} = \mathfrak{F}_{\mathrm{is}}
\]

\[
  \mathfrak{F}_{\mathrm{red}} = \delta\mathfrak{F} - \delta\mathfrak{F}^{*}
\]

\[
  \delta\mathfrak{F} = \underbrace{\mathfrak{F}_{\mathrm{red}} \cup \mathfrak{F}_{\mathrm{lac}}}_{\mathfrak{F}_{\mathrm{int}}}
  \cup \mathfrak{F}_{\mathrm{is}} \cup \partial\mathfrak{F}
\]

\[
  \partial\mathfrak{F} = \underbrace{\partial_0\mathfrak{F}}_{\mathfrak{F}_0 \cap \partial\mathfrak{F}} \amalg \partial'_0\mathfrak{F}
\]

\[
  \delta\mathfrak{F} = \delta\mathfrak{F} = \mathfrak{F}_{\mathrm{red}} \cup \mathfrak{F}_{\mathrm{is}} \cup \mathfrak{F}_{\mathrm{lac}} \cup \{\partial^{*}\mathfrak{F} \ldots
\]

\[
  \partial\mathfrak{F} \cap \mathfrak{F}_{\mathrm{int}} = \partial I \cap \text{\struck{$\partial I$}} (\mathfrak{F}_0 \smallsetminus \mathfrak{F}_{\mathrm{is}})
\]

\[
  \text{\struck{$\partial\mathfrak{F}^{*} \cap \delta\mathfrak{F} = \partial\mathfrak{F} \smallsetminus \partial\mathfrak{F} \cap \partial I$}}
\]

\[
  \text{\struck{$\partial\mathfrak{F} = (\partial\mathfrak{F}^{*} \cap \partial\mathfrak{F}) \amalg \partial\mathfrak{F} \cap \partial I$}}
\]

\[
  \text{\struck{$\partial\mathfrak{F} = (\delta\mathfrak{F}^{*} \cap \partial\mathfrak{F}) \amalg \mathfrak{F}_{\mathrm{is}} \cap \partial I$}}
\]

\[
  \left.
  \begin{aligned}
    \mathfrak{F}_{\mathrm{is}} &= \mathfrak{F}^{*}_{\mathrm{lac}} \\
    \mathfrak{F}_{\mathrm{lac}} &= \mathfrak{F}^{*}_{\mathrm{is}}
  \end{aligned}
  \right\}
\]

\[
  \mathfrak{F}_{\mathrm{red}} = \delta\mathfrak{F} \smallsetminus \delta\mathfrak{F} \cap \delta\mathfrak{F}^{*}
  \qquad \text{NB}\ \mathfrak{F}^{*}_{\mathrm{red}} = \partial I \smallsetminus (\partial I \cap \partial\mathfrak{F}) \ldots
\]

\[
  = \underbrace{(\delta\mathfrak{F} \smallsetminus \delta\mathfrak{F}^{**})}_{\text{sommets évanescents}}
  \amalg \underbrace{(\partial I \cap \partial\mathfrak{F})}_{\text{sommets redondants}}
\]

\[
  \partial^{*}\mathfrak{F} = \partial^{*}\mathfrak{F}^{*} \qquad \text{où} \quad
  \partial^{*}\mathfrak{F} = \partial\mathfrak{F} \smallsetminus \partial\mathfrak{F} \cap \partial I
\]

\[
  \delta\mathfrak{F} = \mathfrak{F}_{\mathrm{red}} \cup \mathfrak{F}_{\mathrm{is}} \cup \mathfrak{F}_{\mathrm{lac}} \cup \partial^{*}\mathfrak{F}
\]

\[
  \widehat{\delta}\mathfrak{F} \overset{\mathrm{def}}{=} \delta\mathfrak{F} \cup \partial I
  = \delta\mathfrak{F} \cup \underbrace{(\partial I \smallsetminus \partial I \cap \delta\mathfrak{F})}_{= \mathfrak{F}^{*}_{\mathrm{sév}}}
\]

\[
  \widehat{\delta}\mathfrak{F} = \mathfrak{F}_{\mathrm{is}} \cup \mathfrak{F}_{\mathrm{lac}} \cup \mathfrak{F}_{\mathrm{év}} \cup \mathfrak{F}_{\mathrm{sév}} \cup \mathfrak{F}^{*}_{\mathrm{sév}} \cup \partial^{*}\mathfrak{F}
\]

\[
  \mathfrak{F}_{\mathrm{is}} = \mathfrak{F}^{*}_{\mathrm{lac}} , \quad
  \mathfrak{F}_{\mathrm{lac}} = \mathfrak{F}^{*}_{\mathrm{is}} , \quad
  \partial^{*}\mathfrak{F} = \partial^{*}\mathfrak{F}^{*} , \quad
  \mathfrak{F}^{*}_{\mathrm{év}} = \mathfrak{F}^{**}_{\mathrm{év}} = \emptyset .
\]

\[
  \mathfrak{F}_{\mathrm{sév}} = \{ s \in \partial I \mid s \notin \delta\mathfrak{F}^{*} \}
  = \partial I \smallsetminus \partial I \cap \delta\mathfrak{F}^{*}
\]

\[
  \mathfrak{F}^{*}_{\mathrm{sév}} = \{ s \in \partial I \mid s \notin \delta\mathfrak{F} \}
  = \partial I - \partial I \cap \delta\mathfrak{F}
\]

\[
  \boxed{\operatorname{cosupp}^{\circ}(\mathfrak{F}) = \operatorname{cosupp}^{\circ}\bigl(\operatorname{Omb}^{\circ}(\mathfrak{F})\bigr) = \operatorname{Omb}^{\circ}(\mathfrak{F}^{*})}
\]

\[
  \mathfrak{F} \longmapsto \operatorname{Omb}^{\circ}(\mathfrak{F}) \qquad \operatorname{Préfig}(\mathfrak{M}) \to \operatorname{Cons}(\mathfrak{M})
\]

\[
  t_0 = \omega_0 < t_1 < \cdots < t_n = \omega_1
\]

\[
  \underset{\substack{\| \\ \omega_0}}{\{t_0\}},\ I_{t_0 t_1},\ t_1,\ I_{t_1, t_2},\ \ldots,\ I_{t_{n-1}, t_n},\ \underset{\substack{\| \\ \omega_n}}{t_n} .
\]

\[
  \underbrace{(\complement\Phi)^{**}}_{\substack{\text{polygone}\\ \text{induit}\\ \text{associé à } \complement\Phi}} \;=\; \Phi^{*}
\]

\[
  \boxed{\operatorname{Cosupp}^{\circ}\Phi \;=\; \operatorname{supp}^{\circ}\operatorname{Omb}^{\circ}\complement\Phi \;=\; \operatorname{supp}^{\circ}\complement\Phi .}
\]

\[
  S_\Phi\subset\mathcal{M}, \qquad S_\Phi=\operatorname{supp}^{\circ}(\Omega) =\operatorname{supp}^{\circ}(\operatorname{Omb}^{\circ}\Phi) .
\]

\[
  \Phi\longmapsto S_\Phi, \qquad \mathfrak{P}(\Phi_T)\longrightarrow \underbrace{\Sigma_{\mathrm{cons}}(\mathcal{M})}_{\substack{\text{ens. des}\\ \text{supports}\\ \text{constructibles}}}
\]

\[
  \operatorname{card}\Phi_T=2n+1
\]

\[
  \underbrace{\operatorname{card}\mathfrak{P}(\Phi_T)}_{\substack{\text{nb de supports}\\ \text{subordonnés à } T}} \;=\; 2^{2n+1}.
\]

\[
  \text{\struck{\ill{}}}\ \operatorname{Omb}^{\circ}(\{I_{\omega_0,\omega_1},\omega_1\})=\mathcal{M}\smallsetminus\omega_0
\]

\[
  (\overbrace{I\smallsetminus\{\omega_0\}}^{J},\ \omega_1)
\]

\[
  \mathcal{M}(J,\omega_1)\ (=\mathcal{M}(I,\partial I)\smallsetminus\{\omega_0\})
\]

\[
  a,\quad J_{<a},\quad J_{a,b}\qquad a<b,\ a,b\in J .
\]

\[
  \mathcal{M}(J,\omega_1)=\{\omega_1,\ J_{<\omega_1}\}
\]

\[
  |J_{<\omega_1}|^{\circ}=\emptyset, \quad\text{mais bien sûr}\quad \mathring{J}_{<\omega_1}\neq\emptyset .
\]

\[
  \text{pour}\qquad \mathcal{M}(I,\partial I)\smallsetminus\{\omega_1\} \overset{\mathrm{def}}{=} \mathcal{M}(J,\omega_0)
\]

\[
  J_{>a}\qquad (a\in J)
\]

\[
  \mathcal{M}(I,\partial I)\smallsetminus\partial I \overset{\mathrm{def}}{=} \mathcal{M}(J) \qquad J=I\smallsetminus\partial I .
\]

\[
  [J]
\]

\[
  \{J_{<a},\ J_{>a}\}\quad\text{si } a\in J
\]

\[
  J_{<a},\ J_{a,b},\ J_{>b}\quad\text{si } a<b,\ a,b\in J .
\]

\[
  (I,\Omega_0,\Omega_1)
\]

\[
  I,\ \text{un ordonné}; \qquad \Omega_0,\Omega_1\subset I
\]

\[
  \partial J \overset{\mathrm{df}}{=} \Omega_0\cup\Omega_1 .
\]

\[
  \mathcal{M}(J,\Omega_0,\Omega_1), \quad\text{où } J,\Omega_0,\Omega_1
\]

\[
  (*)\quad J=\Sigma_\Phi\cap\mathcal{M}_0=\operatorname{Omb}^{\circ}_{I}(\Phi)
\]

\[
  =\Bigl\{s\in I \Bigm| \text{a) } a\leq s\leq b,\ \ \text{b) } s\cup\delta\Phi\in\operatorname{Diag}_I,\ \ \text{c) } s\in\{a,b\}\Rightarrow s\in\Phi^{\circ}\Bigr\}
\]

\[
  |X|^{\circ}\quad (X\in\Phi),
\]

\[
  \Omega_0=\begin{cases}
    \emptyset & \text{si la première strate de $\Phi$ n'a pas d'origine,}\\
              & \text{i.e. est un $I_{<a}$, ou si elle a une origine $a\notin\Phi$}\\
    \{a=\operatorname{or}(\Phi)\} & \text{sinon}
  \end{cases}
\]

\[
  \text{\struck{$\Omega_1=\operatorname{ex}(\Phi)\cap$}}
\]

\[
    x\mathbin{|0|}y \iff \begin{cases}
      x\in\mathcal{M}_i,\ y\in\mathcal{M}_j,\ i\neq j\\
      \text{\textit{ou}}\\
      x,y\in\mathcal{M}_i,\ x\mathbin{|0|}y \text{ au sens de } \mathcal{M}_i .
    \end{cases}
  \]

\[
  \mathcal{F}\subset\mathfrak{P}(\mathcal{M}),\qquad \mathcal{F}=\{A\subset\mathcal{M}\mid \forall i\in\mathcal{M}_i,\ A\cap\mathcal{M}_i\in\mathcal{F}_i\}.
\]

\[
  \mathcal{F}=\prod_{i\in I}\mathcal{F}_i .
\]

\[
  \begin{array}{ccc}
    X & \mathring{\ll} & Y\\
    \triangledown\!\!/ & & \\
    X' & &
  \end{array}
\]

\[
  \begin{array}{ccccc}
    X & \mathring{\ll} & Y & \mathring{\ll} & Z\in\Phi\\
    \triangledown\!\!/ & & \triangledown\!\!/ & & \triangledown\!\!/\\
    X' & \mathring{\ll} & Y' & \mathring{\ll} & Z'\\
     & & \overset{?}{\in} & & \\
     & & \operatorname{Omb}^{\circ}(\Phi) & &
  \end{array}
  \quad ;\ \text{et comme } Z'\in\overline{\Phi},
\]

\[
  \mathcal{M}'=\operatorname{supp}^{\circ}\Phi \quad(\text{un support constructible}).
\]

\[
  \begin{aligned}
    &Y=\text{2-simplexe }(a,b,c)\\
    &X=\text{2-simplexe }(a,b,d)\\
    &X'=\text{1-simplexe }(b,d)\\
    &\text{\struck{$\Phi=\ldots$}}\\
    &Y'=\text{1-simplexe }(b,c)\\
    &\Phi=\{d\}.
  \end{aligned}
  \qquad
  \text{On a}\quad \Phi\mathbin{|0|}X,\ \ \Phi\mathbin{|0|}Y,\ \text{et}\ \Phi\mathbin{|0|}X',
\]

\[
  \mathcal{M}'=\{X\in\mathcal{M}\mid X\mathbin{|0|}\Phi,\ X\not\geq\Phi\} \quad ?
\]

\[
  \begin{array}{ccc}
    X & \mathring{\ll} & Y\\
    \triangledown\!\!/ & & \triangledown\!\!/\\
    X' & \mathring{\ll} & Y'
  \end{array}
  \qquad \text{alors}\quad X,Y,X'\in\mathcal{M}\Rightarrow Y'\in\mathcal{M}.
\]

\[
  \forall X\in\mathcal{M}_c,\ \text{\struck{posant}}\ \text{\add{l'ens.}}\ \Phi_X=\{Y\in\Phi\mid Y\mathbin{\overline{\|}}X\}\ \text{\add{est fini}}
\]

\[
  \text{\struck{, on a $\Phi\smallsetminus\Phi_X$ fini.}}
\]

\[
  \mathcal{M}\subset\mathcal{F}_{\mathrm{\ell f}}(\mathcal{M})
\]

\[
  \{X'\in\widetilde{X}\mid X'\mathbin{\overline{\|}}Y\}
\]

\[
  (\mathcal{M},\trianglelefteq,\mathring{\ll},|0|,\mathcal{F}).
\]

\[
  \mathcal{M}_c=\{X\in\mathcal{M}\mid \forall F\in\mathcal{F},\ \text{l'ens. } \{Y\in\widetilde{F}\mid X\mathbin{\overline{\|}}Y\}\ \text{est fini}\}
\]

\[
  \mathcal{M}_c=\{X\in\mathcal{M}\mid \forall Y\in\mathcal{M},\ \text{l'ens. } \{Y'\in\widetilde{Y}\mid Y'\mathbin{\overline{\|}}X\}\ \text{fini}\},
\]

\[
  (\mathcal{M},\trianglelefteq,\mathring{\ll},|0|,\mathcal{M}_c).
\]

\[
  \operatorname{Figil}(\mathcal{L})=\mathcal{M}
\]

\[
  (\mathcal{A},\trianglelefteq,\mathring{\ll},|0|).
\]

\[
  \Sigma_{\mathcal{A}}\subset\mathfrak{P}(\mathcal{A}),
\]

\[
  S \mathrel{|\circ|} S' \iff \forall X \in S,\ X' \in S',\ \text{on a }
  X \mathrel{|\circ|} X' .
\]

\[
  S \mathrel{|\circ|} S' \iff S \subset \complement S' \iff S' \subset \complement S .
\]

\[
  S \mathrel{|\circ|} S' \Longrightarrow S \cap S' = \varnothing ,
\]

\[
  X^{\circ} \in \Sigma\mathrm{cons}_{\mathcal{A}}
\]

\[
  \operatorname{D\acute{e}p}(X) = \{ X'^{\circ} \mid X' \in \widetilde{X} \}
  \subset \Sigma\mathrm{cons} \subset \Sigma
\]

\[
  X \ll Y \qquad (X, Y \in \mathcal{A})
\]

\[
  \varphi : \widetilde{X} \longrightarrow \widetilde{Y}
\]

\[
  X' \overset{\circ}{\ll} \text{\struck{$\varphi(X')$}}\
  Y' \overset{\text{déf}}{=} \varphi(X') .
\]

\[
  \psi : \operatorname{D\acute{e}p}(X) \longrightarrow \operatorname{D\acute{e}p}(Y)
\]

\[
  \forall S \in \operatorname{D\acute{e}p} X, \quad S \leq \psi(S)
\]

\[
  S \leq T, T', \quad \text{\uncertain{avec}} \quad
  T, T' \in \operatorname{D\acute{e}p}(Y)
\]

\[
  X \ll \text{\struck{\ill{}}}\ Y ,
\]

\[
  \psi : \operatorname{D\acute{e}p}(X) \longrightarrow \operatorname{D\acute{e}p}(Y)
\]

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

\[
  \text{\struck{\ill{}}}\ (\mathcal{A}, \trianglelefteq, \mathrel{|\circ|}) ,
\]

\[
  (\mathcal{A}, \trianglelefteq, \overset{\circ}{\ll}, \mathrel{|\circ|}) \ ?
\]

\[
  X \longmapsto \operatorname{D\acute{e}p}(X) \qquad
  \mathcal{A} \longrightarrow \operatorname{Fig\acute{e}l\,ord}(\Sigma)
\]

\[
  \operatorname{D\acute{e}p}(X) \overset{\circ}{\ll} \operatorname{D\acute{e}p}(Y)
\]

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

\[
  \operatorname{D\acute{e}p}(X) \ll \operatorname{D\acute{e}p}(Y)
\]

\[
  \operatorname{D\acute{e}p} X \overset{\circ}{\ll} \mathfrak{F}
  \leq \text{\struck{$\ll$}}\ \operatorname{D\acute{e}p}(Y)
\]

\[
  \mathfrak{F} = \operatorname{D\acute{e}p}(Y'), \qquad Y' \trianglelefteq Y ,
\]

\[
  S \mathrel{|\circ|} T,\ S' \leq S,\ T' \leq T \Longrightarrow
  S' \mathrel{|\circ|} T'
\]

\[
  (\mathcal{A}, \trianglelefteq, \mathrel{|\circ|})
\]

\[
  \mathcal{A} \longrightarrow
  \text{\struck{$\mathfrak{F}$}}\ \operatorname{Fig\acute{e}l}(\mathfrak{P}(\mathcal{A}))
\]

\[
  X \longmapsto \operatorname{D\acute{e}p} X
  = \bigl( \{ X'^{\circ} \mid X' \trianglelefteq X \} ,
\]

\[
  Y^{\circ} = \{ Z \in \mathcal{A} \mid \forall Z' \in \mathcal{A},\
  Z' \mathrel{|\circ|} Y \Rightarrow Z' \mathrel{|\circ|} Z \} \ ) .
\]

\[
  \Sigma = \Sigma_{\mathcal{A}}
\]

\[
  x \leq y \iff \complement y \leq \complement x
\]

\[
  X \mathrel{|\circ|} Y \iff X \leq \complement Y
  \quad (\iff Y \leq \complement X) .
\]

\[
  X \mathrel{|\circ|} Y \Longrightarrow X \wedge \text{\struck{\ill{}}}\ Y = 0_{\Sigma}
\]

\[
  X \mathrel{|\circ|} Y \ \text{est relation symétrique}
\]

\[
  X \mathrel{|\circ|} X \iff X = 0_{\Sigma}
\]

\[
  \Sigma \subset \mathfrak{P}(\mathcal{A})
\]

\[
  X \mathrel{|\circ|} X \Longrightarrow X \in \mathcal{A}_{00}
\]

\[
  \Sigma_{0} = \{ 0_{\Sigma} \} = \Sigma_{00} .
\]

\[
  \operatorname{cosupp}(A) = \{ Z \in \Sigma \mid \forall X \in A,\
  X \mathrel{|\circ|} Z \}
\]

\[
  = \Bigl\{ Z \in \Sigma ,\ Z \leq \inf_{X \in A} \complement X
  = \complement \operatorname{Sup} A \Bigr\}
\]

\[
  = \widetilde{\complement \operatorname{Sup} A} .
\]

\[
  \operatorname{supp} A \overset{\text{déf}}{=}
  \operatorname{cosupp} \operatorname{cosupp} A
  = \complement\bigl(\operatorname{Sup} \widetilde{\complement(\xi)}\bigr)
  = \widetilde{\xi}
\]

\[
  \mathfrak{F} \subset \Sigma
\]

\[
  \text{\struck{$(\mathfrak{F}, \rho) \trianglelefteq \mathfrak{F}'$}} \qquad
  \operatorname{Atspat}(\Sigma) \subset \operatorname{Atens}(\Sigma)
\]

\[
  \mathcal{A} \longrightarrow \Sigma
\]

\[
  X = (\mathfrak{F}, \rho) \longmapsto \text{plus grand élément de } \mathfrak{F},
\]

\[
  \Sigma_{\mathcal{A}} \longrightarrow \Sigma_{\Sigma} = \Sigma .
\]

\[
  \Sigma \longrightarrow \mathcal{A}
\]

\[
  X \longmapsto \text{\struck{$\{X\}$}}\ (\{X\}, \rho_{X}) \qquad
  \rho_{X} \ \text{la relation d'ordre (unique) sur } \{X\}
\]

\[
  \Sigma \longrightarrow \mathcal{A} \longrightarrow \Sigma
\]

\[
  \Sigma_{\mathcal{A}} \simeq \Sigma_{\Sigma} \simeq \Sigma
\]

\[
  \mathcal{A} \longrightarrow \operatorname{Atspat}(\Sigma_{\mathcal{A}})
\]

\[
  \mathcal{A} \longrightarrow \operatorname{Atspat}(\Sigma_{\mathcal{A}})
  \xrightarrow{\ \sim\ } \operatorname{Atspat}(\Sigma) = \mathcal{A}
\]

\[
  \mathcal{A} \longrightarrow \operatorname{Atspat}(\Sigma)
\]

\[
  X \longmapsto \operatorname{D\acute{e}p}(X) ,
\]

\[
  X \mathrel{|\circ|} Y \iff
  \operatorname{D\acute{e}p}(X) \mathrel{|\circ|} \operatorname{D\acute{e}p} Y .
\]

\[
  \mathcal{A} \subset \operatorname{Atspat}(\Sigma) ,
\]

\[
  \Sigma' = \{ X^{\circ} \mid X \in \mathcal{A} \} \subset \Sigma .
\]

\[
  S = \operatorname{Sup} \Sigma'_{\leq S} .
\]

\[
  \Sigma \simeq \Sigma_{\mathcal{A}}
\]

\[
  \mathcal{A} \xrightarrow{\ \operatorname{D\acute{e}p}\ }
  \operatorname{Atspat}(\Sigma_{\mathcal{A}})
\]

\[
  \mathcal{A} \xrightarrow{\ \operatorname{D\acute{e}p}\ }
  \operatorname{Figspat\acute{e}l}(\Sigma_{\mathcal{A}})
\]

\[
  \mathcal{A} \xrightarrow{\ \alpha\ } \Sigma
  \xrightarrow{\ \beta\ } \mathfrak{P}(\mathcal{A})
\]

\[
  S \longmapsto \{ X \in \mathcal{A} \mid X^{\circ} \leq S \}
\]

\[
  X = (\mathfrak{F}, \rho) \longmapsto X^{\circ} =
  \text{plus grand él. de } \mathfrak{F} \text{ pour } \rho
\]

\[
  \beta(\alpha(X)) = \{ Z \in \mathcal{A} \mid Z^{\circ} \leq X^{\circ} \}
  \subset \operatorname{supp}^{\circ}_{\mathcal{A}} X
\]

\[
  \alpha\beta(S) = \{ X^{\circ} \mid X \in \mathcal{A},\ \text{t.q. }
  X^{\circ} \leq S \} \subset \operatorname{supp}^{\circ}_{\Sigma}(S)
\]

\[
  \Sigma_{\mathcal{A}} \quad \text{et} \quad \Sigma_{\Sigma} .
\]

\[
  \Sigma_{\alpha} : \Sigma_{\mathcal{A}} \longrightarrow \Sigma_{\Sigma}
  \overset{\operatorname{supp}^{\circ}_{\Sigma}}{\rightleftarrows} \Sigma ,
\]

\[
  \mathcal{A} \xrightarrow{\ \operatorname{D\acute{e}p}^{!}_{\mathcal{A}}\ }
  \text{\struck{\ill{}}}\ \mathfrak{P}(\Sigma_{\mathcal{A}})
  \longrightarrow \mathfrak{P}(\Sigma)
\]

\[
  X \longmapsto \{ \operatorname{supp}^{\circ}_{\mathcal{A}}(X') \mid
  X' \in \widetilde{X} \} \overset{?}{=}
\]

\[
  X \mathrel{|\circ|} X \Longrightarrow \forall Y \in E,\
  X \mathrel{|\circ|} Y .
\]

\[
  E \xrightarrow{\ \varphi\ } \mathfrak{P}(E')
\]

\[
  \forall x, y \in E, \ \text{et } x' \in \varphi(x),\ y' \in \varphi(y),
  \quad x \mathrel{|\circ|} y \Longrightarrow x' \mathrel{|\circ|} y'
\]

\[ E \xrightarrow{\ \alpha\ } \mathfrak{P}(E') \]

\[ \bar\alpha : \mathfrak{P}(E) \longrightarrow \mathfrak{P}(E'), \qquad A \longmapsto \bigcup_{x \in A} \alpha(x), \]

\[ \Sigma_E \xhookrightarrow{\ \mathrm{inc}\ } \mathfrak{P}(E) \xrightarrow{\ \bar\alpha\ } \mathfrak{P}(E') \xrightarrow{\ \operatorname{supp}^{\circ}_{E'}\ } \Sigma_{E'} ; \]

\[ \alpha_{*} : \Sigma_E \longrightarrow \Sigma_{E'}, \qquad \alpha_{*}(S) = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(S)\bigr). \]

\[ (*) \qquad \boxed{\ \alpha_{*}\bigl(\operatorname{supp}^{\circ}_{E} A\bigr) = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(A)\bigr)\ } \]

\[ \text{\struck{$\alpha_{*}(\bar A) = \operatorname{supp}^{\circ}_{E'}(\bar\alpha(\bar A)) \supset \operatorname{supp}^{\circ}_{E'}(\bar\alpha(A))$}} \]

\[ \text{\struck{$\operatorname{supp}^{\circ}_{E'}(\bar\alpha(\bar A)) \subset \operatorname{supp}^{\circ}_{E'}(\bar\alpha(A))$,}} \]

\[ \text{\struck{$x' \mathrel{|o|} \bar\alpha(A) \Longrightarrow x' \mathrel{|o|} \bar\alpha(\bar A)$~?}} \]

\[ (**) \qquad \alpha_{*}\bigl(\operatorname{cosupp}^{\circ}_{E}(B)\bigr) = \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(B)\bigr) \]

\begin{align*}
\alpha_{*}\bigl(\operatorname{supp}^{\circ}(B)\bigr) &= \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(\operatorname{cosupp}^{\circ}_{E}(A))\bigr) \\
&= \operatorname{cosupp}^{\circ}_{E'}\Bigl(\operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(\ \text{---}\ )\bigr)\Bigr)
\end{align*}

\begin{align*}
\uncertain{dans}\ \operatorname{supp}^{\circ}_{E}(A) &= \operatorname{cosupp}^{\circ}_{E'}\bigl(\operatorname{cosupp}^{\circ}_{E'}(\bar\alpha(A))\bigr) \\
&= \operatorname{supp}^{\circ}_{E'}(\bar\alpha(A)) \qquad \text{cqfd.}
\end{align*}

\begin{align*}
\alpha_{*}(\complement B) &= \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(B)\bigr) \\
&= \operatorname{cosupp}^{\circ}_{E'}\bigl(\underbrace{\operatorname{supp}^{\circ}_{E'}(\bar\alpha(B))}_{\overset{\text{déf}}{=}\ \alpha_{*}(B)}\bigr) = \complement\,\alpha_{*}(B),
\end{align*}

\[ \boxed{\ \alpha_{*}(\complement B) = \complement\,\alpha_{*}(B)\ } \]

\[ C = \operatorname{cosupp}^{\circ} B = \{\, x \in E \mid x \mathrel{|o|} B \,\}~; \]

\[ \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(C)\bigr) = \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(B)\bigr) \qquad \bigl(C = \operatorname{cosupp}^{\circ}_{E}(B)\bigr). \]

\[ \bar\alpha(C) \subset \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(B)\bigr), \]

\[ \bar\alpha(C) \mathrel{|o|_{E'}} \bar\alpha(B) \]

\[ B \mathrel{|o|_{E}} C, \]

\[ \operatorname{cosupp}^{\circ}_{E'}\bigl(\bar\alpha(B)\bigr) \subset \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(C)\bigr), \]

\[ x' \in \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(C)\bigr), \]

\[ x', y' \in E', \quad x' \mathrel{|o|} \bar\alpha(B), \quad y' \mathrel{|o|} \bar\alpha(C) \ \Longrightarrow\ x' \mathrel{|o|} y'. \]

\[ E' \xrightarrow{\ \beta\ } \mathfrak{P}(E), \]

\[ \mathfrak{P}(E') \xrightarrow{\ \bar\beta\ } \mathfrak{P}(E) \]

\[ (*) \qquad \forall x' \in E', \quad \bar\alpha\bigl(\beta(x')\bigr) \subset \operatorname{supp}^{\circ}_{E'}(\{x'\}). \]

\[ x' \mathrel{|o|} \bar\alpha(B) \Longleftrightarrow \bar\alpha\bigl(\beta(x')\bigr) \mathrel{|o|} \bar\alpha(B) \]

\[ \beta(x') \mathrel{|o|} B. \]

\[ \beta(x') \mathrel{|o|} B, \qquad \underbrace{\beta(y') \mathrel{|o|} C = \operatorname{cosupp}^{\circ}(B)}_{\text{i.e. } \beta(y') \subset \operatorname{supp}^{\circ}(B)} \]

\[ \beta(x') \mathrel{|o|} \beta(y'), \qquad \text{d'où} \]

\[ \bar\alpha\bigl(\beta(x')\bigr) \mathrel{|o|} \bar\alpha\bigl(\beta(y')\bigr) \]

\[ \operatorname{supp}^{\circ}_{E'}\Bigl(\bar\alpha\bigl(\beta(x')\bigr)\Bigr) = \operatorname{supp}^{\circ}_{E'}\bigl(\{x'\}\bigr). \]

\begin{gather*}
\text{Soit } \beta = {}^{t}\alpha^{*} \text{ défini par} \\
\alpha^{*}(x') = \bigl\{\, x \in E \mid \alpha(x) \subset \operatorname{supp}^{\circ}_{E'}(\{x'\}) \,\bigr\}
\end{gather*}

\[ \bar\alpha\bigl(\alpha^{*}(x')\bigr) \subset \operatorname{supp}^{\circ}_{E'}(\{x'\}), \]

\[ \text{(i)} \qquad \boxed{\ \operatorname{supp}^{\circ}_{E'}\Bigl(\bar\alpha\bigl(\alpha^{*}(x')\bigr)\Bigr) = \operatorname{supp}^{\circ}_{E'}(\{x'\})\ } \]

\[ \text{(ii)} \qquad \boxed{\ \text{si } x, y \in E,\quad x \mathrel{|o|} y \Longleftrightarrow \bar\alpha(x) \mathrel{|o|} \bar\alpha(y)\ } \]

\[ \bar\alpha(A) \mathrel{|o|} \bar\alpha(B) \Longleftrightarrow A \mathrel{|o|} B. \]

\[ \alpha_{*} : \Sigma_E \xrightarrow{\ \sim\ } \Sigma_{E'} \]

\[ \underbrace{\alpha_{*}\operatorname{supp}^{\circ}_{E}(A)}_{\overset{\text{déf}}{=}\ \operatorname{supp}^{\circ}_{E'}\bar\alpha(\operatorname{supp}^{\circ}_{E}(A))} = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(A)\bigr) \]

\[ \bar\alpha\bigl(\operatorname{supp}^{\circ}_{E}(A)\bigr) \supset \bar\alpha(A) \]

\[ \underbrace{\alpha_{*}\bigl(\operatorname{supp}^{\circ}_{E}(A)\bigr)}_{\operatorname{supp}^{\circ}_{E'}(\bar\alpha(\operatorname{supp}^{\circ}_{E}(A)))} \overset{?}{\subset} \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(A)\bigr) \]

\begin{align*}
&\text{Si } B \subset \operatorname{supp}^{\circ}_{E}(A) \text{ i.e. } \forall x \in E,\ x \mathrel{|o|} A \Rightarrow x \mathrel{|o|} B, \\
&\text{peut-on en conclure que} \\
&\overset{?}{\Bigl\{}\ \bar\alpha(B) \subset \text{\struck{$\bar\alpha(\operatorname{supp}$}}\ \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(A)\bigr) \text{ i.e. } \forall x' \in E', \\
&\qquad x' \mathrel{|o|} \bar\alpha(A) \Rightarrow x' \mathrel{|o|} \bar\alpha(B)~?
\end{align*}

\[ \beta : E' \longrightarrow \mathfrak{P}(E) \]

\begin{align*}
(*) \qquad & x \mathrel{|o|} \beta(x') \Longleftrightarrow \alpha(x) \mathrel{|o|} x', \\
&\text{d'où, pour } A \subset E,\ B' \subset E' \\
(**) \qquad & A \mathrel{|o|} \bar\beta(B') \Longleftrightarrow \bar\alpha(A) \mathrel{|o|} B'
\end{align*}

\[ \text{\struck{$x \mathrel{|o|} \beta(x) \Rightarrow$}} \qquad \beta(x') \mathrel{|o|} A \overset{?}{\Longrightarrow} \beta(x') \mathrel{|o|} B \]

\[ \gamma(x') = \{\, x \in E \mid \alpha(x) \mathrel{|o|} x' \,\}, \]

\[ x \mathrel{|o|} \beta(x') \Longleftrightarrow x \in \gamma(x') \qquad \text{i.e.} \]

\[ \gamma(x') = \operatorname{cosupp}^{\circ}\bigl(\beta(x')\bigr) \qquad \text{\struck{\ill{}}} \]

\begin{align*}
1^{\circ})\quad & \gamma(x') \in \Sigma_E \\
2^{\circ})\quad & \text{\struck{$\beta$}}\ \operatorname{supp}^{\circ}\beta(x') = \complement\,\gamma(x') \quad \bigl(= \operatorname{cosupp}^{\circ}(\gamma(x'))\bigr)
\end{align*}

\[ \beta(x') = \operatorname{cosupp}^{\circ}\bigl(\gamma(x')\bigr) = \bigl\{\, x \in E \mid \forall y \in E,\ \alpha(y) \mathrel{|o|} x' \Longrightarrow x \mathrel{|o|} y \,\bigr\}~; \]

\[ \text{\struck{$\alpha_{*} : \Sigma_E \longrightarrow \Sigma_{E'}$}} \]

\[ \alpha_{*} : \mathfrak{P}(E) \longrightarrow \mathfrak{P}(E') \]

\[ \alpha_{*} : \Sigma_E \longrightarrow \Sigma_{E'} \]

\[ \alpha : E \longrightarrow \Sigma_{E'}, \qquad x \longmapsto \alpha_{*}\bigl(\underbrace{\operatorname{supp}^{\circ}_{E}(\{x\})}_{s(x)}\bigr) \]

\[ \alpha_{*}(S) = \mathop{\mathrm{Sup}}\limits_{x \in S\ (\text{dans } \Sigma_{E'})} \bigl(\underbrace{\alpha(x)}_{\alpha_{*}(s(x))}\bigr) = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(S)\bigr) \]

\[ S = \Bigl[\mathop{\mathrm{Sup}}\limits_{x \in S\ (\text{dans } \Sigma_E)}\Bigr](s_x). \]

\[ \text{\struck{$\alpha(x) \ni$}}\qquad x' \in \alpha(x) \Longleftrightarrow x \in \beta(x') \]

\[ x' \mathrel{|o|} \alpha(x) \Longleftrightarrow x \mathrel{|o|} \beta(x') \]

\[ \alpha_{*} : \Sigma^{*}_{A} \longrightarrow \Sigma^{*}_{A'} \]

\[ \alpha : E \longrightarrow \mathfrak{P}(E'), \]

\[ \text{\struck{i.e.\ \ill{}}}\qquad \alpha_{*}(S) = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(S)\bigr). \]

\[ (*) \qquad \alpha_{*}(S) \mathrel{|o|} S' \Longleftrightarrow S \mathrel{|o|} \beta_{*}(S') \]

\[ (**) \qquad \alpha(x) \mathrel{|o|} x' \Longleftrightarrow x \mathrel{|o|} \beta(x'). \]

\[ \beta_{*}(S') = \operatorname{supp}^{\circ}_{E}\bigl(\bar\beta(S')\bigr) \]

\[ \bar\beta(x') = \{\, x \in E \mid \alpha(x) \mathrel{|o|} x' \,\} \subset E~; \]

\[ \bar\beta(x') \in \Sigma_E. \]

\[ \beta_{*}(x') = \complement\bigl(\underbrace{\bar\beta(x')}_{\in\,\Sigma_E}\bigr) = \operatorname{cosupp}^{\circ}_{E}\bigl(\bar\beta(x')\bigr) \]

\begin{align*}
\operatorname{supp}^{\circ}_{E}\beta(x') &= \complement\bigl(\bar\beta(x')\bigr) \\
\text{ou aussi}\qquad \operatorname{cosupp}^{\circ}\bigl(\beta(x')\bigr) &= \bar\beta(x').
\end{align*}

\[ \alpha_{*} \longmapsto {}^{t}\alpha_{*} : \operatorname{Corrdis}(\underline{E}, \underline{E}') \longrightarrow \operatorname{Corrdis}(\underline{E}', \underline{E}) \]

\[ \begin{cases} {}^{t}({}^{t}\alpha_{*}) = \alpha_{*} \\ {}^{t}({}^{t}\beta_{*}) = \beta_{*} \end{cases} \]

\[ (***) \qquad \underbrace{\alpha_{*}\bigl(\operatorname{supp}^{\circ}_{E}(A)\bigr)}_{\overset{\text{déf}}{=}\ \operatorname{supp}^{\circ}_{E'}(\bar\alpha(\operatorname{supp}^{\circ}_{E}(A)))} = \operatorname{supp}^{\circ}_{E'}\bigl(\bar\alpha(A)\bigr) \]

\[ A = \bigl\{\, \{x, y\} \in \mathfrak{P}_2(E) \mid x \mathrel{\overline{|o|}} y \,\bigr\}. \]

\[ \complement S \overset{\mathrm{df}}{=} \operatorname{cosupp}^{\circ}_{E} A \qquad (A \in \Sigma_E \subset \mathfrak{P}(E)), \]

\[ x \longmapsto \sigma(x) = \operatorname{supp}^{\circ}_{E}(\{x\}) : E \xrightarrow{\ \sigma\ } \Sigma_E \]

\[ x \mathrel{|o|} y \Longleftrightarrow x \neq y \]

\[ \Sigma_E = \mathfrak{P}(E). \]

\begin{align*}
\operatorname{cosupp}^{\circ}_{E}(A) &= \operatorname{cosupp}^{\circ}_{E}(A \cup E_0) \supset E_0, \\
\operatorname{supp}^{\circ}_{E}(A) &= \operatorname{supp}^{\circ}_{E}(A \cup E_0) \supset E_0,
\end{align*}

\begin{align*}
\operatorname{cosupp}^{\circ}_{E}(A) &= \operatorname{cosupp}^{\circ}(A^{*}) \\
\operatorname{supp}^{\circ}_{E}(A) &= \operatorname{supp}^{\circ}(A^{*})
\end{align*}

\[ A^{*} = A \setminus E_0 \cap A. \]

\[ \Sigma_E \xrightarrow{\ \sim\ } \Sigma_{E^{*}}, \qquad
\begin{array}{l} S \longmapsto S \setminus E_0 \\ S^{*} \amalg E_0 \longleftarrow S^{*} \end{array} \]

\[ x \leq y \Longleftrightarrow \complement y \leq \complement x. \]

\[ \left[\begin{array}{ll} \text{\emph{Cor 1}} & \complement(\operatorname*{Sup}_i x_i) = \operatorname*{Inf}_i \complement x_i, \\
 & \complement(\operatorname*{Inf}_i x_i) = \operatorname*{Sup}_i \complement x_i. \\
\text{\emph{Cor.\ 2}} & \complement 0 = 1,\quad \complement 1 = 0 \end{array}\right. \]

\[ \text{\struck{$x \mathrel{|o|} y \overset{\mathrm{def}}{\Longleftrightarrow} x \leq \complement y$}} \qquad \text{\struck{($\Longleftrightarrow y \leq \complement x$ par Spat.\ 2).}} \]

\[ x \wedge \complement x = 0_\Sigma \]

\[ x \vee \complement x = 1_\Sigma. \]

\[ x \mathrel{|o|} y \overset{\mathrm{def}}{\Longleftrightarrow} x \leq \complement y \qquad (\Longleftrightarrow y \leq \complement x, \text{ par Spat 2}). \]

\[ x \mathrel{|o|} y,\quad x' \leq x,\quad y' \leq y \Longrightarrow x' \mathrel{|o|} y'. \]

\[ A \cap \operatorname{cosupp}^{\circ}_{E}(A) \subset E_0, \]

\[ A \cap \operatorname{cosupp}^{\circ}_{E}(A) = E_0. \]

\[ E \longrightarrow \Sigma_{\underline{E}}. \]

\[ \sigma : \Sigma \longrightarrow \Sigma_{\Sigma} \]

\[ \sigma^{-1} : S \longmapsto \operatorname{Sup} S. \]

\[ (*) \qquad \Sigma \longrightarrow \Sigma_{\underline{\Sigma}} \qquad (\text{où } \underline{\Sigma} = (\Sigma, |o|)) \]

\[ \sigma_E : E \longrightarrow \Sigma_E, \]

\[ \alpha : E \longrightarrow \Sigma \]

\[
  S = \operatorname{Sup}_{x \in \beta(S)} \sigma x
\]

\[
  \alpha : \Sigma_E \longrightarrow \Sigma,
\]

\[
  \alpha(A) = \operatorname{Sup}_{x \in A} \sigma(x).
\]

\[
  \beta(S) = \operatorname{Cosup}^{\circ}(\beta(S'))
\]

\[
  x \in \beta(S) \iff x \mathrel{|\circ|} \beta(S') \quad \text{i.e.} \quad \sigma(x) \mathrel{|\circ|} \underbrace{\sigma(\beta(S'))}_{S'_0}
\]

\[
  \Sigma \xrightarrow{\ \beta\ } \Sigma_E
\]

\[
  \alpha\beta = \mathrm{id}_{\Sigma} .
\]

\[
  \beta\alpha = \mathrm{id}_{\Sigma_E} ,
\]

\[
  \alpha(A) \leq \complement\alpha(A') ,
\]

\[
  \beta(\alpha(A)) \leq \beta\complement\alpha(A') = \complement\beta(\alpha(A')) , \quad \text{i.e.} \ldots
\]

\[
  \beta\alpha(A) \mathrel{|\circ|} \beta\alpha(A') ,
  \qquad
  \beta\alpha(A) \geq A , \quad \beta\alpha(A') \geq A' ,
\]

\[
  A \leq B \leq \complement B' \leq \complement A' = A ,
\]

\[
  \alpha(A) = \operatorname{Sup}_{x \in A} \underbrace{\alpha(\sigma_E(x))}_{\sigma(x)} = \operatorname{Sup}_{x \in A} \sigma(x) ,
\]

\[
    S \longmapsto S \cap E' , \qquad \Sigma_E \longrightarrow \mathfrak{P}(E')
  \]

\[
    \psi : \Sigma_{E'} \longrightarrow \Sigma_E , \qquad S' \longmapsto \operatorname{Sup}^{\circ}_E(S')
  \]

\[
  \xi = \operatorname{Sup} S' .
\]

\[
  \text{\struck{$S' \mathrel{|\circ|} y$}} \quad S' \mathrel{|\circ|} y \Longrightarrow \xi \mathrel{|\circ|} y .
\]

\[
  S' \mathrel{|\circ|} y \iff \xi \mathrel{|\circ|} y .
\]

\[
  \forall\, y \in E, \quad
  \underbrace{S' \mathrel{|\circ|} y}_{\text{i.e. } y \leq \operatorname{Inf}_{\xi' \in S'} \complement(\xi')}
  \Longrightarrow
  \underbrace{\xi \mathrel{|\circ|} y}_{\text{i.e. } y \leq \complement(\xi)}
\]

\[
  \operatorname{Inf}_{\xi' \in S'} \complement(\xi') \leq \complement(\xi) ,
\]

\[
  \xi \leq \operatorname{Sup}_{\xi' \in S'} \xi' ,
\]

\[
  S' = \operatorname{Sup} \sigma(\beta(S)) \leq S .
\]

\[
  \forall\, x, y \in E, \quad E'_{\leq x} \mathrel{|\circ|} E'_{\leq y} \Longrightarrow x \mathrel{|\circ|} y .
\]

\[
  \Sigma_{E'} \underset{\varphi}{\overset{\psi}{\rightleftarrows}} \Sigma_E
\]

\[
  \psi(A') = \operatorname{Sup}^{\circ}_E(A') , \qquad \varphi(A) = A \cap E' .
\]

\[
  \alpha : E \longrightarrow \mathfrak{P}(E') \qquad (x \longmapsto R(x))
\]

\[
  \alpha_{*} \text{ ou } \overline{\alpha} : \mathfrak{P}(E) \longrightarrow \mathfrak{P}(E') , \qquad A \longmapsto \bigcup_{x \in A} \alpha(x) .
\]

\[
  A = \operatorname{Sup}_{x \in A} \{x\} \quad \text{d'où} \quad \varphi(A) = \operatorname{Sup}_{x \in A} \underbrace{\varphi(\{x\})}_{\text{\uncertain{déf.} } \alpha(x)} .
\]

\[
  \beta_{*} \text{ ou } \overline{\beta} : \mathfrak{P}(E') \longrightarrow \mathfrak{P}(E) ,
\]

\[
  \forall\, A \in \mathfrak{P}(E),\ A' \in \mathfrak{P}(E') \quad \text{on a} \quad
  \alpha_{*}(A) \mathrel{|\circ|} A' \iff A \mathrel{|\circ|} \beta_{*}(A') .
\]

\[
  \text{\struck{$\forall\, x \in$}}\ \forall\, x' \in A', \ \text{on a } x' \notin \overline{\alpha}(A) \ \text{ i.e. } \nexists\, x \in A
\]

\[
  \text{on a encore} \quad \forall\, x \in A,\ x' \in A' \quad \text{on a } (x, x') \notin R \ \text{ i.e. } (x, x') \in \overline{R}
\]

\[
  \alpha : E \to \mathfrak{P}(E') \ \text{ou}\ \overline{\alpha} : \mathfrak{P}(E) \to \mathfrak{P}(E') ,
  \qquad
  \beta : E' \to \mathfrak{P}(E) \ \text{ou}\ \overline{\beta} : \mathfrak{P}(E') \to \mathfrak{P}(E) .
\]

\[
    x \mathrel{|\circ|} \beta(x') \iff \alpha(x) \mathrel{|\circ|} x' .
  \]

\[
    A \mathrel{|\circ|} \overline{\beta}(A') \iff \overline{\alpha}(A) \mathrel{|\circ|} A' .
  \]

\[
    \alpha_{*}(A) = \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) , \qquad
    \beta_{*}(A') = \operatorname{Sup}^{\circ}_{E}(\overline{\beta}(A')) ,
  \]

\[
    \text{\struck{$\alpha_{*}(A)$}}\ A \mathrel{|\circ|} \beta_{*}(A') \iff \alpha_{*}(A) \mathrel{|\circ|} A' .
  \]

\[
  \text{\struck{$\alpha_{*}(\operatorname{Sup}^{\circ}_E(A)) = \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A))$,}}
  \quad
  \text{\struck{$\beta_{*}(\operatorname{Sup}^{\circ}_E(A')) = \operatorname{Sup}^{\circ}_E(\overline{\beta}(B))$}}
\]

\[
  \Sigma_E \underset{\beta_{*}}{\overset{\alpha_{*}}{\rightleftarrows}} \Sigma_{E'}
\]

\[
  \alpha'(x) = \operatorname{Sup}^{\circ}_{E'} \alpha(x) , \qquad
  \beta'(x) = \operatorname{Sup}^{\circ}_{E} \beta(x) .
\]

\[
  A \mathrel{|\circ|} \overline{\beta}(A') \iff x \mathrel{|\circ|} \overline{\beta}(x') \quad \forall\, x \in A,\ x' \in A'
\]

\[
  \alpha(A) \mathrel{|\circ|} A' \iff \alpha(x) \mathrel{|\circ|} x' .
\]

\[
  (\ast) \qquad
  \underbrace{\operatorname{Sup}^{\circ}_E(A)}_{\overline{A}} \mathrel{|\circ|} \operatorname{Sup}^{\circ}_E(\overline{\beta}(A'))
  \iff
  \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) \mathrel{|\circ|} \underbrace{\operatorname{Sup}^{\circ} A'}_{\overline{A'}}
\]

\[
  \text{\struck{$\operatorname{Sup}^{\circ}_E(\overline{\beta}(A')) = \operatorname{Sup}^{\circ}_E$}}
\]

\[
  (\ast\ast) \quad
  \left\{
  \begin{aligned}
    \operatorname{Sup}^{\circ}_E(\overline{\beta}(A')) &= \operatorname{Sup}^{\circ}_E(\overline{\beta}(\overline{A'})) \quad \bigl(\overset{\text{déf}}{=} \beta_{*}(\overline{A'})\bigr) \\
    \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) &= \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(\overline{A})) \quad \bigl(\overset{\text{déf}}{=} \alpha_{*}(\overline{A})\bigr)
  \end{aligned}
  \right.
\]

\[
  \overline{A} \mathrel{|\circ|} \beta_{*}(\overline{A'}) \iff \alpha_{*}(\overline{A}) \mathrel{|\circ|} \overline{A'}
\]

\[
  \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) \subset \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(\overline{A})) ,
\]

\[
  A' \mathrel{|\circ|} \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) \iff A' \mathrel{|\circ|} \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(\overline{A}))
\]

\[
  \text{\struck{$A' \mathrel{|\circ|} \overline{\alpha}(A) \overset{?}{\Longrightarrow} \beta_{*}(A') \mathrel{|\circ|} A$}}
\]

\[
  \operatorname{Sup}^{\circ}_{E'} \alpha(x) = \operatorname{Sup}^{\circ}_{E'} \overline{\alpha}\bigl(\operatorname{Sup}^{\circ}_E(x)\bigr) \overset{\text{déf}}{=} \alpha_{*}\bigl(\operatorname{Sup}^{\circ}_E(x)\bigr) ,
\]

\[
  \boxed{\ \alpha_{*}\bigl(\operatorname{Sup}^{\circ}_E(x)\bigr) = \underbrace{\operatorname{Sup}^{\circ}_{E'}(\alpha(x))}_{\overset{\text{déf}}{=}\ \alpha'(x)}\ }
\]

\[
  \boxed{\ \alpha_{*}(S) = \operatorname{Sup}_{x \in S}^{(\Sigma_{E'})} \alpha'(x) \quad (= \alpha'_{*}(S))\ }
\]

\[
  \left\{
  \begin{aligned}
    \alpha' &: E \to \Sigma_{E'} \\
    \beta' &: E' \to \Sigma_E
  \end{aligned}
  \right.
  \qquad
  \left\{
  \begin{aligned}
    \alpha_{*} &: \Sigma_E \to \Sigma_{E'} \\
    \beta_{*} &: \Sigma_{E'} \to \Sigma_E
  \end{aligned}
  \right.
\]

\[
  \alpha : E \longrightarrow \Sigma_{E'} ,
\]

\[
  \alpha_{*} : \Sigma_E \to \Sigma_{E'} , \qquad \alpha_{*}(S) = \operatorname{Sup}^{\Sigma_{E'}}_{x \in S} \alpha(x)
\]

\[
  \beta : E' \longrightarrow \Sigma_E ,
\]

\[
  \beta_{*} : \Sigma_{E'} \to \Sigma_E , \qquad \beta_{*}(S') = \operatorname{Sup}^{\Sigma_E}_{x' \in S'} \beta(x')
\]

\[
  \alpha' : E \to \Sigma_{E'} , \quad \alpha'(x) \overset{\text{déf}}{=} \operatorname{Sup}^{\circ}_{E'}(\alpha(x))
\]

\[
  \alpha_{*} : \Sigma_E \to \Sigma_{E'} , \quad
  \alpha_{*}(S) = \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(S)) = \operatorname{Sup}^{\Sigma_{E'}}_{x \in S} \alpha'(x) .
\]

\[
    \beta : E' \longrightarrow \mathfrak{P}(E)
  \]

\[
    \text{(i.e. } \operatorname{Sup}^{\circ}_{E'}\bigl(\alpha(\operatorname{Sup}^{\circ}_E(x))\bigr)\text{)} = \operatorname{Sup}^{\circ}_{E'}(\alpha(x))
  \]

\[
    \alpha_{*}\bigl(\underbrace{\operatorname{Sup}^{\circ}_E(A)}_{\overline{A}}\bigr) = \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A))
  \]

\[
    S = \{x \in E \mid \alpha(x) \subset S'\}
  \]

\[
    \beta(x') = \{x \in E \mid \alpha(x) \mathrel{|\circ|} x'\}
  \]

\[
  \text{(i)} \Longrightarrow \text{(ii)} \Longrightarrow \text{(iii)} \Longrightarrow \text{(iv)} \Longrightarrow \text{(i)} ,
\]

\[
  \overline{\alpha}(S) \subset S' \Longrightarrow \overline{\alpha}(\overline{S}) \subset S' ,
\]

\[
  \beta(x') = \complement\, \gamma(x') \quad \ldots
\]

\[
  x \mathrel{|\circ|} \beta(x') \iff \alpha(x) \mathrel{|\circ|} x'
\]

\[
  \text{i.e.} \quad \text{\struck{$\operatorname{Sup}^{\circ}_E(x) \subset$}} \quad x \in \complement\,\beta(x') = \gamma(x')
\]

\[
  S = \operatorname{Sup}^{\Sigma_E} S_i = \operatorname{Sup}^{\circ}_E\Bigl(\bigcup S_i\Bigr) . \quad \text{On a} \ldots
\]

\[
  \alpha_{*}(S) = \alpha_{*}\Bigl(\operatorname{Sup}^{\circ}_E\Bigl(\bigcup S_i\Bigr)\Bigr)
  = \operatorname{Sup}^{\circ}_{E'}\Bigl(\underbrace{\overline{\alpha}\Bigl(\bigcup S_i\Bigr)}_{\bigcup_i \overline{\alpha}(S_i)}\Bigr)
\]

\[
  = \operatorname{Sup}^{\circ}_{E'} \bigcup_i \underbrace{\operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(S_i))}_{\alpha_{*}(S_i)} = \operatorname{Sup}_i \alpha_{*}(S_i) , \quad \text{ok}
\]

\[
  \alpha_{*}\bigl(\operatorname{Sup}^{\circ}_E(A)\bigr) \overset{?}{=} \operatorname{Sup}^{\circ}_{E'}(\overline{\alpha}(A)) .
\]

\[
  \operatorname{Sup}^{\circ}_E(A) = \operatorname{Sup}^{\Sigma_E}_{x \in A} \sigma_E(x)
\]

\[
  \alpha_{*}\bigl(\operatorname{Sup}^{\circ}_E(A)\bigr) = \operatorname{Sup}^{\Sigma_{E'}}_{x \in A} \alpha_{*}(\sigma_E(x))
\]

\[
  \operatorname{Sup}^{\circ}_{E'}\bigl(\underbrace{\overline{\alpha}(A)}_{\bigcup_{x \in A} \alpha(x)}\bigr) = \operatorname{Sup}^{\Sigma_{E'}}_{x \in A} \operatorname{Sup}^{\circ}_{E'} \alpha(x)
\]

\[
  \alpha_{*}\bigl(\operatorname{Sup}^{\circ}_E(x)\bigr) = \operatorname{Sup}^{\circ}_{E'}(\alpha(x))
\]

\[
  \operatorname{Corr}(\Sigma, \Sigma')
\]

\[
  \alpha_{*} : \Sigma \longrightarrow \Sigma'
\]

\[
  \operatorname{Corr}(\underline{E}, \underline{E}')
\]

\[
  \alpha : E \longrightarrow \text{\struck{$\Sigma_{E'}$}}\ \mathfrak{P}(E')
\]

\[
  \operatorname{Corr}^{*}(\underline{E}, \underline{E}')
\]

\[
  \left\{
  \begin{aligned}
    &\forall\, x \in E, \ R(x) \in \Sigma_{E'} \\
    &\forall\, x' \in E', \ {}^{t}R(x') \in \Sigma_E .
  \end{aligned}
  \right.
\]

\[
  \operatorname{Corr}(\Sigma, \Sigma') \xrightarrow[\sim]{\ \varphi\ } \operatorname{Corr}(\underline{E}, \underline{E}') \xrightarrow[\sim]{\ \psi\ } \operatorname{Corr}^{*}(\underline{E}, \underline{E}')
\]

\[
  \left\{
  \begin{aligned}
    \varphi(\alpha_{*}) &= \alpha_{*} \circ \sigma_E : \ E \xrightarrow{\ \sigma_E\ } \Sigma_E \xrightarrow{\ \alpha_{*}\ } \Sigma_{E'} \\
    \psi(\alpha) &= \{(x, x') \in E \times E' \mid \alpha(x) \mathrel{|\circ|} x'\} \\
    &\phantom{=}\ \text{i.e. } x' \in \operatorname{Cosup}^{\circ}_{E'}(\alpha(x))
  \end{aligned}
  \right.
\]

\[
  E \xrightarrow{\ \alpha\ } \Sigma_{E'} \xrightarrow{\ \complement_{E'}\ } \Sigma_{E'} \subset \mathfrak{P}(E') \text{).}
\]

\[
  \alpha_{*}(S) = \operatorname{Sup}^{\Sigma_{E'}}_{x \in S} \underbrace{\alpha(x)}_{\overset{\text{déf}}{=}\ \alpha_{*}(\operatorname{Sup}^{\circ}_E(x))} \qquad (S \in \Sigma_E) .
\]

\[
  S = \operatorname{Sup}_{x \in S} \operatorname{Sup}^{\circ}_E(x)
\]

\[
  \overline{\alpha}\bigl(\text{\struck{$\operatorname{Sup}^{\circ}_E$}}\ \sigma_E(x)\bigr) \text{ \struck{$\in$} et } \alpha(x) \text{ ont même } \operatorname{Sup}^{\circ}_{E'} .
\]

\[
  \alpha_{*}\sigma_E(y) \leq \alpha_{*}\sigma_E(x)
  \quad \text{i.e.} \quad
  \underbrace{\alpha(y) \leq \alpha(x)}_{\text{dans } \Sigma_{E'}} ,
\]

\[
  \operatorname{Sup}^{\circ}_{E'}\bigl(\overline{\alpha}(\sigma_E(x))\bigr) = \ldots\ \alpha(x)
\]

\[
  \underline{E} \longrightarrow \underline{E}'
\]

\[
  \beta \underset{\text{dis}}{\circ} \alpha = \text{\struck{$\beta \underset{\text{ens}}{\circ} \alpha$}}\ \operatorname{Sup}^{\circ}_{E''}\bigl(\beta \underset{\text{ens}}{\circ} \alpha\bigr) .
\]

\[
  \text{\struck{\ill{}}}\ \sigma_{\underline{E}} = \operatorname{Sup}^{\circ}_{\underline{E}} \circ \mathrm{id}_{\underline{E}} .
\]

\[
  (x, x') \mathrel{|\circ|} (y, y') \overset{\text{déf}}{\iff} x \mathrel{|\circ|} x' \ \text{ou}\ y \mathrel{|\circ|} y' .
\]

\[
  f \le g \;\overset{\text{déf}}{\iff}\; \forall x \in \Sigma,\ f(x) \le g(x) .
\]

\[
  \bigl(\operatorname{Sup}_i f_i\bigr)(x) = \operatorname{Sup}_i f_i(x) \qquad \forall x \in \Sigma .
\]

\[
  \mathrm{Corr}(\Sigma_E, \Sigma') \simeq \Sigma'^{E},
\]

\[
  (\complement f)(x) = \complement f(x) \qquad \forall x \in E ,
\]

\[
  f \mathbin{|\circ|} g \iff \forall x \in E,\ f(x) \mathbin{|\circ|} g(x) ,
\]

\[
  (\complement f)(A) \ne \complement\bigl(f(A)\bigr) \qquad \bigl(A \in \Sigma_E = \mathfrak{P}(E)\bigr)
\]

\[
  f \mathbin{|\circ|} g \not\Longrightarrow f(A) \mathbin{|\circ|} g(A) .
\]

\[
  \Gamma = f' \circ {}^{t}f ,
\]

\[
  \alpha_*\colon \mathfrak{P}(E) \to \mathfrak{P}(E')
\]

\[
    S \mathbin{|\circ|} T \iff \alpha(S) \mathbin{|\circ|} \alpha(T) ;
  \]

\[
    \alpha_*(S) = \emptyset \Longrightarrow S = \emptyset ;
  \]

\[
  0 < b < c < 1 \bigr) .
\]

\[
    \complement(\alpha) \wedge (\gamma \vee \alpha) = \gamma, \quad
    \complement(\beta)(\gamma \vee \alpha) = \alpha .
  \]

\[
  \alpha = b, \quad \beta = c \wedge \complement b, \quad \gamma = \complement c ,
\]

\[
  \text{\struck{$A \in \mathfrak{t}$}} \quad
  E \in \mathfrak{P}^3(I) \;\Bigm|\;
  \left\{
  \begin{array}{l}
    i, i' \in E \text{ ou } i, i' \notin E \text{ ou} \\
    E = \{i\} \text{ ou } E = \{i, j, k\}
  \end{array}
  \right\}
\]

\[
  a', b, c, \qquad b \vee c = A, \quad c \vee a' = B, \quad a' \vee b = C ;
\]

\[
  \bigvee_i e_i = 1_\Sigma .
\]

\[
    e_J = \operatorname*{Sup}_{i \in J} e_i ,
  \]

\[
    \mathfrak{P}(I) \longrightarrow \Sigma, \qquad J \longmapsto e_J
  \]

\[
  \bigl\{ \{e_i\}, e_0 \bigr\}
\]

\[
  \complement e_{I'} = e_{I''} \vee e_0
\]

\[
  e_{I'} = e \wedge \complement e_{I''} .
\]

\[
  \Sigma_{e_*} = \Sigma_{(e_i)} ,
\]

\[
  \text{alors} \quad e' = (\complement e) \wedge (e \vee e') .
\]

\[
  A, B \in \mathrm{Figspatdisc}(\Sigma) ,
\]

\[
  B \ll A \iff \forall b \in B,\ \exists a \in A \text{ avec } b \le a
\]

\[
  B_a = \{ b \in B \mid b \le a \} ,
\]

\[
    \widehat{B} = B \cup \{ a'' \mid a \in A \text{ t.q. } a'' \ne 0 \}
  \]

\[
  B = \{ X^\circ \mid X \in G \}, \qquad A = \{ X^\circ \mid X \in F \} ,
\]

\[
  F_0 \ll F_1 \ll \dots \ll F_n
\]

\[
  A_0 \ll A_1 \ll \dots \ll A_n .
\]

\[
  A_0 \ll A_1 \ll \dots \ll A_n ,
\]

\[
  A = \bigcup_{1 \le i \le n} A_i
\]

\[
  A''_i = \{ a''_i \mid a_i \in A_i \text{ t.q. } a''_i \ne 0_\Sigma \} .
\]

\[
  A''_0 = A_0 , \qquad
  A = \bigcup_{0 \le i \le n} A''_i .
\]

\[
  F_0 \ll F_1 \ll \cdots \ll F_n ,
\]

\[
  A_0 \ll A_1 \ll \cdots \ll A_n
\]

\[
  \operatorname{Figélspatcomm}(\Sigma)
\]

\[
  e = f \vee f' \text{ avec } f \mathrel{|{\circ}|} f', \qquad f = g \vee g' \text{ avec } g \mathrel{|{\circ}|} g',
\]

\[
  \operatorname{Omb}^{\circ}(\mathcal{F}) = \{ X \in \mathcal{A} \mid X \mathrel{\mathring{\ll}} \mathcal{F} \}
\]

\[
  \operatorname{Supp}^{\circ}(S) = \{ X \in \mathcal{A} \mid X \mathrel{|{\circ}|} S \}.
\]

\[
  \{ X \in \mathcal{A} \mid X \mathrel{|{\circ}|} \mathcal{F} \text{ et } X \between \mathcal{F} \}
\]

\[
  X \mathrel{\mathring{\ll}} Y' \trianglelefteq Y \quad \text{avec } X, Y \in \mathcal{A}',
\]

\[
  \forall Z \in \mathcal{F}, \quad Y' \between Z, \text{ et } Y' \mathrel{|{\circ}|} Z \text{ i.e. } Y' \neq Z .
\]

\[
  f^{*} : \operatorname{Figures}(E) \longrightarrow \operatorname{Figures}(E')
\]

\[
  g_{*} : \text{\struck{$\operatorname{Figel}$}}\ \mathcal{A} \longrightarrow \mathcal{A}', \quad \text{\uncertain{alors} injective.}
\]

\[
  f : \mathcal{A} \longrightarrow \hat{\mathcal{A}}' := \mathcal{A}' \amalg \{\emptyset\}
  \qquad \bigl(\hookrightarrow \mathcal{F}(\mathcal{A}')\bigr)
\]

\[
  X' \trianglelefteq Y', \quad X' \mathrel{\mathring{\ll}} Y', \quad X' \mathrel{|{\circ}|} Y'
  \quad \text{pour } X', Y' \in \mathcal{A}' \amalg \{\emptyset\},
\]

\[
  \operatorname{Ker} f \overset{\text{déf}}{=} \{ X \in \mathcal{A} \mid f(X) = \emptyset \}.
\]

\[
  \mathcal{A} \xrightarrow{\;p\;} \mathcal{A}/N
\]

\[
  \mathcal{A} \xrightarrow{\;f\;} \mathcal{A}'
\]

\[
  \mathcal{A} \amalg \{\emptyset\} \xrightarrow{\;\hat{f}\;} \mathcal{A}' \amalg \{\emptyset\}
\]

\[
  \mathcal{A} = \operatorname{Figelann}(E), \qquad N = \{ X \in \mathcal{A} \mid |X| \subset E \setminus E' \},
\]

\[
  \mathcal{A}/N = \operatorname{Figelann}(E'),
\]

\[
  G \ll F
\]

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

\[
  Y \mathrel{\mathring{\ll}} X .
\]

\[
  \operatorname{Supp}^{\circ} \varphi^{-1}(X) = X^{\circ}
\]

\[
  \text{(i)} \Rightarrow \text{(ii)} \Rightarrow \text{(iii)} \Rightarrow \text{(iv)}
\]