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It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  All that remains is to see how this operation on coefficient values in <math>\mathbb{B}</math> induces the corresponding operations on sets and terms.
 
It is clear that these operations are isomorphic, amounting to the same operation of type <math>\mathbb{B} \times \mathbb{B} \to \mathbb{B}.</math>  All that remains is to see how this operation on coefficient values in <math>\mathbb{B}</math> induces the corresponding operations on sets and terms.
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The term <math>\mathit{l}^\mathrm{w}\!</math> determines a selection of individuals from the universe of discourse <math>X\!</math> that may be computed by means of the corresponding operation on coefficient matrices.  If the terms <math>\mathit{l}\!</math> and <math>\mathrm{w}\!</math> are represented by the matrices <math>\mathfrak{L} = \operatorname{Mat}(\mathit{l})</math> and <math>\mathfrak{W} = \operatorname{Mat}(\mathrm{w}),</math> respectively, then the operation on terms that produces the term <math>\mathit{l}^\mathrm{w}\!</math> must be represented by a corresponding operation on matrices, say, <math>\mathfrak{L}^\mathfrak{W} = \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})},</math> that produces the matrix <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}).</math>  In other words, the involution operation on matrices must be defined in such a way that the following equations hold:
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{| align="center" cellspacing="6" width="90%"
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| height="60" | <math>\mathfrak{L}^\mathfrak{W} ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \operatorname{Mat}(\mathit{l}^\mathrm{w})</math>
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{| align="center" cellspacing="6" width="90%"
 
{| align="center" cellspacing="6" width="90%"
| height="60" | <math>\operatorname{Mat}(\mathit{l}^\mathrm{w}) ~=~ \operatorname{Mat}(\mathit{l})^{\operatorname{Mat}(\mathrm{w})} ~=~ \mathfrak{L}^\mathfrak{W}</math>
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| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
 
| height="60" | <math>(\mathfrak{L}^\mathfrak{W})_{a} ~=~ \prod_{x \in X} \mathfrak{L}_{ax}^{\mathfrak{W}_{x}}</math>
 
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