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Exercise 99 - Chapter 2

edited June 2018 in Exercises

Let \( \mathcal{V} = (V, \le, I, \otimes) \) be a quantale. Use Eq. (2.97) and Proposition 2.84 to prove the following.

  1. Show that for any sets \(X\) and \(Y\) and \(V\)-matrix \(M : X \times Y \rightarrow V \), one has \( I_X * M = M \).

  2. Prove the associative law: for any matrices \( M : W \times X \rightarrow V, N : X \times Y \rightarrow V \), and \( P : Y \times Z \rightarrow V \), one has \( (M * N) * P = M * (N * P) \).

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Equation 2.97

$$ ( M * N )(x, z) := \bigvee_{y \in Y} M(x, y) \otimes N(y, z) $$ Proposition 2.84

Suppose \( \mathcal{V} = (V, \le, I, \otimes) \) is a symmetric monoidal preorder, and that it is closed. Then

1) For every \( v \in V \), the monotone map \( − \otimes v : (V, \le) \rightarrow (V, \le) \) is left adjoint to \( v \multimap − : (V, \le) \rightarrow (V, \le) \).

2) For any element \( v \in V \) and set of elements \( A \subseteq V \), we have $$ \left( v \otimes \bigvee_{a \in A} a \right) \cong \bigvee_{a \in A} ( v \otimes a ) $$ 3) For any \( v, w \in V \), we have \( v \otimes (v \multimap w) \le w \).

4) For any \( v \in V \), we have \( v \cong (I \multimap v) \).

5) For any \( u, v, w \in V \), we have \( (u \multimap v) \otimes (v \multimap w) \le (u \multimap w) \).

Comments

  • 1.

    Definition of identity \(V\)-matrix (page 65 (75 of the pdf) of Seven Sketches):\[I_x(x,y)=\begin{cases}I\quad\mathrm{if}\,x=y\\0\quad\textrm{if}\,x\neq y\end{cases}\]

    1. By equation 2.97, \((I_X* M)(x_2,y):=\bigvee_{x_1\in X}I_X(x_2,x_1)\otimes M(x_1,y)\). By the definition of identity, \(\bigvee_{x_1\in X}I_X(x_2,x_1)\otimes M(x_1,y)=I\otimes M(x_2,y)\bigvee_{x_1\in X\\x_1\neq x_2}0\otimes M(x_1,y)\) \(=I\otimes M(x_2,y)\vee0\) \(=I\otimes M(x_2,y)\) \(=M(x_2,y)\). Thus, \((I_X* M)(x_2,y)=M(x_2,y)\). Since this equation holds for all \(x_2\in X\) and \(y\in Y\), one can conclude \(I_X* M=M\). (Note that \(0\otimes a=0\) by Prop. 2.84 2) and symmetry.)

    2. By equation 2.97, \(((M* N)* P)(w,z)=\bigvee_{y\in Y}(M* N)(w,y)\otimes P(y,z)\) \(=\bigvee_{y\in Y}\left(\bigvee_{x\in X}M(w,x)\otimes N(x,y)\right)\otimes P(y,z)\). By Prop. 2.84 2) and associativity of \(\otimes\), \(\bigvee_{y\in Y}\left(\bigvee_{x\in X}M(w,x)\otimes N(x,y)\right)\otimes P(y,z)\) \(=\bigvee_{y\in Y}\bigvee_{x\in X}M(w,x)\otimes N(x,y)\otimes P(y,z)\). Setting \(v=\bigvee_{b\in B}b\), one can use Prop. 2.84 2) to prove that suprema commute. By commutativity of suprema, \(\bigvee_{y\in Y}\bigvee_{x\in X}M(w,x)\otimes N(x,y)\otimes P(y,z)\) \(=\bigvee_{x\in X}\bigvee_{y\in Y}M(w,x)\otimes N(x,y)\otimes P(y,z)\). By Prop. 2.84 2) and associativity of \(\otimes\), \(\bigvee_{x\in X}\bigvee_{y\in Y}M(w,x)\otimes N(x,y)\otimes P(y,z)\) \(=\bigvee_{x\in X}M(w,x)\left(\bigvee_{y\in Y}N(x,y)\otimes P(y,z)\right)\). By equation 2.97, \(\bigvee_{x\in X}M(w,x)\left(\bigvee_{y\in Y}N(x,y)\otimes P(y,z)\right)\) \(=\bigvee_{x\in X}M(w,x)\otimes(N* P)(x,z)\) \(=(M* (N* P))(w,z)\). So, \(((M* N)* P)(w,z)=(M* (N* P))(w,z)\). Since this equation holds for all \(w\in W\) and \(z\in Z\), one can conclude that \((M* N)* P=M* (N* P)\).

    Comment Source:Definition of identity \\(V\\)-matrix (page 65 (75 of the pdf) of Seven Sketches):\\[I_x(x,y)=\begin{cases}I\quad\mathrm{if}\,x=y\\\\0\quad\textrm{if}\,x\neq y\end{cases}\\] 1. By equation 2.97, \\((I_X* M)(x_2,y):=\bigvee_{x_1\in X}I_X(x_2,x_1)\otimes M(x_1,y)\\). By the definition of identity, \\(\bigvee_{x_1\in X}I_X(x_2,x_1)\otimes M(x_1,y)=I\otimes M(x_2,y)\bigvee_{x_1\in X\\\\x_1\neq x_2}0\otimes M(x_1,y)\\) \\(=I\otimes M(x_2,y)\vee0\\) \\(=I\otimes M(x_2,y)\\) \\(=M(x_2,y)\\). Thus, \\((I_X* M)(x_2,y)=M(x_2,y)\\). Since this equation holds for all \\(x_2\in X\\) and \\(y\in Y\\), one can conclude \\(I_X* M=M\\). (Note that \\(0\otimes a=0\\) by Prop. 2.84 2) and symmetry.) 2. By equation 2.97, \\(((M* N)* P)(w,z)=\bigvee_{y\in Y}(M* N)(w,y)\otimes P(y,z)\\) \\(=\bigvee_{y\in Y}\left(\bigvee_{x\in X}M(w,x)\otimes N(x,y)\right)\otimes P(y,z)\\). By Prop. 2.84 2) and associativity of \\(\otimes\\), \\(\bigvee_{y\in Y}\left(\bigvee_{x\in X}M(w,x)\otimes N(x,y)\right)\otimes P(y,z)\\) \\(=\bigvee_{y\in Y}\bigvee_{x\in X}M(w,x)\otimes N(x,y)\otimes P(y,z)\\). Setting \\(v=\bigvee_{b\in B}b\\), one can use Prop. 2.84 2) to prove that suprema commute. By commutativity of suprema, \\(\bigvee_{y\in Y}\bigvee_{x\in X}M(w,x)\otimes N(x,y)\otimes P(y,z)\\) \\(=\bigvee_{x\in X}\bigvee_{y\in Y}M(w,x)\otimes N(x,y)\otimes P(y,z)\\). By Prop. 2.84 2) and associativity of \\(\otimes\\), \\(\bigvee_{x\in X}\bigvee_{y\in Y}M(w,x)\otimes N(x,y)\otimes P(y,z)\\) \\(=\bigvee_{x\in X}M(w,x)\left(\bigvee_{y\in Y}N(x,y)\otimes P(y,z)\right)\\). By equation 2.97, \\(\bigvee_{x\in X}M(w,x)\left(\bigvee_{y\in Y}N(x,y)\otimes P(y,z)\right)\\) \\(=\bigvee_{x\in X}M(w,x)\otimes(N* P)(x,z)\\) \\(=(M* (N* P))(w,z)\\). So, \\(((M* N)* P)(w,z)=(M* (N* P))(w,z)\\). Since this equation holds for all \\(w\in W\\) and \\(z\in Z\\), one can conclude that \\((M* N)* P=M* (N* P)\\).
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