\[
  C_{\mathrm{mod}}(T) = \{t \in \mathcal{L} \smallsetminus T \mid \{T \cup \{t\}\} \text{ modérée}\}
\]

\[
  S \cup T = \mathcal{L},
\]

\[
  \mathfrak{S}_0 = \mathfrak{P}_2(\mathcal{L}) \Longrightarrow
  \left\{
  \begin{aligned}
    &\mathrm{Modf}(\mathcal{L}) = \mathfrak{P}_f(\mathcal{L}) \\
    &C_T = \mathcal{L} \smallsetminus T \quad \text{si } T \in \mathfrak{P}_f(\mathcal{L})
  \end{aligned}
  \right.
\]

\[
  R_T = \text{relation d'équivalence discrète dans } \mathcal{L} \smallsetminus T,
\]

\[
  \dot{A} = \emptyset
\]

\[
  \underbrace{\dot{A}}_{\subset\, S} \cap\, T \subset \underbrace{\dot{B}}_{\subset\, T}
\]

\[
  \mathfrak{S}_0 \subset \mathfrak{P}_2(\mathcal{L})
\]

\[
  \mathrm{Modf}(\mathcal{L}) = \text{parties finies tot. ordonnées}
\]

\[
  \underbrace{]-\infty, t_1[}_{t_1},\ \underbrace{]t_1, t_2[}_{t_1,\, t_2},\ \ldots,\ \underbrace{]t_{n-1}, t_n[}_{t_{n-1},\, t_n},\ \underbrace{]t_n, +\infty[}_{t_n}
\]

\[
  [s,t] \quad (s \leq t), \qquad ]-\infty, s] \text{ ou } [s, +\infty[, \qquad \mathcal{L} = ]-\infty, +\infty[
\]

\[
  \partial \mathcal{L} = \{x \in \mathcal{L} \mid x \text{ est plus petit élément ou plus grand élément}\}
\]

\[
  \begin{aligned}
    \mathfrak{S}_0(\mathcal{L}) = \mathrm{Drap}_2(\mathcal{L}) &= \{\varepsilon \in \mathfrak{P}_2(\mathcal{L}) \mid \varepsilon \text{ totalement ordonné}\} \\
    &= \{\{x,y\} \mid x < y\}
  \end{aligned}
\]

\[
  \text{Déf. :=} \qquad \mathrm{Modf}(\mathcal{L}) = \mathrm{Drap}(\mathcal{L}) = \{S \subset \mathfrak{P}_f(\mathcal{L}) \mid S \text{ tot. ord.}\}
\]

\[
  C(S) = \{t \in \mathcal{L} \mid t \notin S,\ S \cup \{t\} \text{ tot. ord.}\}
\]

\[
  \mathcal{L}_{<s_1},\ [s_1, s_2],\ \ldots,\ [s_{n-1}, s_n],\ \mathcal{L}_{>s_n}
\]

\[
  \underbrace{\{s_1\}}_{\text{si comp. existe}},\ \{s_1, s_2\},\ \ldots,\ \{s_{n-1}, s_n\},\ \underbrace{\{s_n\}}_{\text{si comp. existe}}
\]

\[
  \mathcal{R}(a,b,c) \qquad \text{$b$ est \textbf{strictement entre} $a$ et $c$}
\]

\[
  \left.
  \begin{aligned}
    \mathcal{L}_{\geq b} &= \{x \in \mathcal{L} \mid x = b \text{ ou } b \text{ entre } a \text{ et } x\} \\
    \mathcal{L}_{\leq a} &= \{x \in \mathcal{L} \mid x = a \text{ ou } a \text{ entre } b \text{ et } x\} \\
    ]a, b[ &= \{x \in \mathcal{L} \mid x \text{ entre } a \text{ et } b\}
  \end{aligned}
  \right\}
\]

\[
  x \in \mathcal{L}_{\leq a},\quad y \in \mathcal{L}_{\geq b} \quad \text{et} \quad x \leq u \leq v \leq y
\]

\[
  x \leq u \leq v \leq y
\]

\[
  \left\{
  \begin{aligned}
    &\mathcal{R}(a,b) &&\{a,b\} \in \mathrm{Drap}_2(\mathcal{L}) \\
    &\mathcal{R}(a,b,c) &&b \text{ est str. entre } a \text{ et } c
  \end{aligned}
  \right.
\]

\[
  C(\varepsilon) = \{t \in \mathcal{L}_{\notin \varepsilon} \mid \{\varepsilon, t\} \in \mathrm{Drap}_3(\mathcal{L})\}
\]

\[
  \mathcal{R}(a,b,c) \ : \ b \text{ strict entre } a \text{ et } c.
\]

\[
  \mathcal{J}_{abc} = \text{composante de } C(\{a,b\}),
\]

\[
  \begin{aligned}
    X_{\{a,b\}} &= (\text{comp. de } C(\{a,b\}) \text{ contenant } c) = \mathcal{L}_{>b} \\
    X_{\{b,c\}} &= (\text{comp. de } C(\{b,c\}) \text{ contenant } a) = \mathcal{L}_{<b} \\
    X_{\{c,a\}} &= (\text{comp. de } C(\{c,a\}) \text{ contenant } b) = ]a, c[
  \end{aligned}
\]

\[
  \left.
  \begin{aligned}
    X_{\{a,b\}} \cap X_{\{b,c\}} &= \emptyset \\
    X_{\{a,b\}} \cap X_{\{c,a\}} &= ]b, c[ \neq \emptyset \\
    X_{\{b,c\}} \cap X_{\{c,a\}} &= ]a, b[ \neq \emptyset
  \end{aligned}
  \right\} \text{divisibilité !}
\]

\[
  X_{\varepsilon} \cap X_{\varepsilon'} = \emptyset.
\]

\[
    \left\{
    \begin{aligned}
      &T_i \cap T_j = \emptyset \\
      &T_i \text{ et } T_j \text{ en pos. rel. mod.}
    \end{aligned}
    \right.
  \]

\[
  \boxed{(X \cap Y)^{\cdot} \subset \dot{X} \cup \dot{Y}}
\]

\[
  \boxed{(X \cup_{\mathrm{mod}} Y)^{\cdot} \subset \dot{X} \cup \dot{Y}}
\]

\[
  \Bigl(\bigcup_{\mathrm{mod},\, i} X_i\Bigr)^{\cdot} \subset \bigcup_i \dot{X}_i
\]

\[
  [a,b] \cup_{\mathrm{mod}} [b,c] = [a,c]
\]

\[
  [a,b] \cup [b,c] \neq [a,c]
\]

\[
  \underbrace{\mathcal{L}_{\leq b} \cup_{\mathrm{mod}} \mathcal{L}_{\geq b}}_{\mathcal{L}} = \mathcal{L}_{\leq b} \cup \mathcal{L}_{\geq b}
\]

\[
  C_{\mathrm{mod}}(Y, X) = \{x \in X \mid x \notin Y,\ x \text{ en pos. rel. mod. avec } Y\}
\]

\[
  C_{\mathrm{mod}}(Y,X) = C_{\mathrm{mod}}(F, Z)
\]

\[
  \begin{array}{ll}
    L_{<a},\ L_{>a},\ ]a,b[ & (a < b) \\
    \complement L_{\geq a},\ \complement L_{\leq a},\ [a,b] & (a \leq b)
  \end{array}
\]

\[
  \overline{]a,b[} = [a,b] \ ?
\]

\[
  \overline{L_{<a}} = L_{\leq a} \ ? \quad \text{(si $a$ n'est pas plus petit élément ($a \neq 0_L$))}
\]

\[
  \overline{L_{>a}} = L_{\geq a} \ ? \quad \text{(si $a$ n'est pas plus grand élément ?)}
\]

\[
  \partial [a,b] = \{a,b\}, \quad \partial L_{\leq a} = \{a\}, \quad \partial L_{\geq a} = \{a\}, \quad \partial L = \emptyset
\]

\[
  \partial T = \text{bord absolu de } T = \Bigl\{ t \in T \;\Big|\; t \text{ est un plus petit ou plus grand élément de } T \Bigr\}
\]

\[
  \begin{array}{ll}
    \partial \{a\} = \{a\} & \\
    \partial [a,b] = \{a,b\} & \text{si } a \leq b \\
    \partial L_{\geq a} = \partial L_{\leq a} = \{a\} & \forall a \in L \\
    \partial L = \partial L \ \text{(sic)} &
  \end{array}
\]

\[
  \dot{T} \cap T \subset \partial T \subset (\dot{T} \cap T) \cup \partial L
\]

\[
  \boxed{\partial T = (\dot{T} \cap T) \sqcup (T \cap \partial L)}
\]

\[
  \left\{
  \begin{array}{ll}
    ]a,b],\ [a,b[,\ ]a,b[ & (a < b) \\
    L_{<a} & (a \neq 0_L) \\
    L_{>a} & (a \neq \omega_L) \\
    L &
  \end{array}
  \right.
\]

\[
  T' = T \smallsetminus \alpha \qquad (T \text{ tronçon fermé, } \alpha \subset \partial T)
\]

\[
  \partial T' = \partial T
\]

\[
  \partial T' = \dot{T}' \sqcup \partial L \ \ldots
\]

\[
  \partial (T \cap T') \subset \partial T \cup \partial T'
\]

\[
  T \prec T'
\]

\[
  \forall t \in T,\ t' \in T' \Longrightarrow t < t'
\]

\[
  T \ll T' \iff T \prec T', \text{ et position relative propre}
\]

\[
  \iff \exists x \in L \text{ tel que } T \subset L_{\leq x},\ T' \subset L_{>x}
\]

\[
  (a \in T \text{ ou } b \in T') \Longrightarrow a \neq b
\]

\[
  T \prec S \prec T'
\]

\[
  T \subset L_{<x}, \quad T' \subset L_{>x} \qquad \text{(cf. b))}
\]

\[
  (*) \qquad S \prec T, \quad S \text{ et } T \text{ se raccordent}, \quad S \text{ sans minorant}
\]

\[
  X_{\Phi} = \bigcup_{i \in \Phi} T_i \qquad \text{(NB réunion disjointe)}
\]

\[
  \Phi \longmapsto X_{\Phi}, \qquad \underbrace{\mathrm{Drap}^{*}_{\ll}(\mathrm{Tr}(L))}_{\text{drapeaux de } \mathrm{Tr}(L), \text{ y inclus}\ \ldots} \longrightarrow \mathfrak{P}(L)
\]

\[
  X_{\Phi} = \bigcup T_i
\]

\[
  \mathrm{Fr}(X_{\Phi}) = \underbrace{\Bigl( \bigcup \partial T_i \Bigr)}_{\partial \mathcal{U}, \text{ drapeau de } L} \smallsetminus \varepsilon
\]

\[
  \varepsilon = \Bigl\{ t \in \partial L \cap \mathcal{U} \;\Big|\; \{t\} \notin \Phi \Bigr\}
\]

\[
  X' = C_{\mathrm{mod}}(\mathcal{U}) = \bigl\{ x \in L \smallsetminus \mathcal{U} \;\big|\; x \text{ est en pos. rel. mod. p.r. à } \mathcal{U} \bigr\}
\]

\[
  \{ T_1 \ll T_2 < \cdots \ll T_n \}
\]

\[
  n' \in \{ n-1, n, n+1 \}
\]

\[
  \bigl| \mathrm{Card}\, \pi_0(X) - \mathrm{Card}\, \pi_0(X') \bigr| \leq 1
\]

\[
  \begin{array}{ll}
    T_1 \text{ non minoré, } T_n \text{ non majoré} & : n' = n-1 \\
    T_1 \text{ minoré et } T_n \text{ non majoré, ou l'inverse} & : n' = n \\
    T_1 \text{ minoré et } T_n \text{ majoré} & : n' = n+1 .
  \end{array}
\]

\[
  \{X', X\} \text{ modérée}, \quad X' \wedge X = \emptyset, \quad X' \vee X = 1
\]

\[
  (X \wedge Y)' = X' \vee Y', \quad (X \vee Y)' = X' \wedge Y'
\]

\[
  \left\{
  \begin{array}{l}
    \mathrm{Fr}\, X \overset{\mathrm{def}}{=} \overline{X} \wedge \overline{X'} \\
    (\mathrm{Fr}\, X)' = X^{\circ} \vee X'^{\circ}
  \end{array}
  \right.
\]

\[
  \begin{array}{ll}
    X + Y = (X \vee Y) \wedge (X \wedge Y)' & \qquad X \vee Y = X + Y - XY \\
    X \cdot Y = X \wedge Y & \qquad X \wedge Y = X \cdot Y \\
    0 = \text{plus petit él.}, \ 1 = \text{plus grd él.} & \qquad X \leq Y \iff X \cdot Y = X
  \end{array}
\]

\[
  \mathcal{M}_A = \mathcal{M}_{\leq A} = \{ Y \in \mathcal{M} \mid Y \subset A \}
\]

\[
  \overline{X}^{(\mathcal{M}_A)} = \overline{X}^{\mathcal{M}} \wedge A
\]

\[
  \overline{X \vee Y}^{\mathcal{M}_A} = \underbrace{\overline{X \vee Y}^{\mathcal{M}}}_{} \wedge A = (\overline{X} \vee \overline{Y}) \wedge A = \underbrace{(\overline{X} \wedge A)}_{\overline{X}^{A}} \vee (\overline{Y} \wedge A) = \overline{X}^{A} \vee \overline{Y}^{A}
\]

\[
  A = X_{\Phi} = \bigcup T_i, \qquad \Phi \in \mathrm{Drap}^{*}_{\ll}(\mathrm{Tr}(L))
\]

\[
  A = \bigvee_i T_i \text{ dans } \mathcal{M}, \quad \struck{\text{et}} \text{ et } T_i \wedge T_j = \emptyset \text{ si } i \neq j,
\]

\[
  \underbrace{\overline{T_i} \cap A}_{} = T_i ,
\]

\[
  X = X_{\Phi} = \bigcup_{i \in I} T_i ,
\]

\[
  S = \bigcup_{\alpha} \dot{X}_{\alpha}
\]

\[
  \underset{\text{si } t_1 \notin \partial L}{[L_{<t_1}]},\ \{t_1\},\ ]t_1, t_2[,\ \{t_2\},\ ]t_1, t_3[,\ \{t_3\}, \ldots, \{t_n\},\ \underset{\text{si } t_n \notin \partial L}{[L_{>t_n}]}
\]

\[
  S \subset \underset{\displaystyle \mathrm{Drap}_2(L)}{\overset{\|}{\mathfrak{P}_2(L)}}
\]

\[
  \complement_{\mathrm{mod}}\, \varepsilon = \bigl\{ x \in L \;\big|\; \{a,x\}, \{b,x\} \in S \bigr\}
\]