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### 6 Boolean algebras

#### 6.1 Properties

##### 6.1-1 IsBiHeytingAlgebroid
 ‣ IsBiHeytingAlgebroid( C ) ( property )

Returns: true or false

The property of C being a bi-Heyting algebroid.

##### 6.1-2 IsBiHeytingAlgebra
 ‣ IsBiHeytingAlgebra( C ) ( property )

Returns: true or false

The property of C being a bi-Heyting algebra.

##### 6.1-3 IsBooleanAlgebroid
 ‣ IsBooleanAlgebroid( C ) ( property )

Returns: true or false

The property of C being a Boolean algebroid.

##### 6.1-4 IsBooleanAlgebra
 ‣ IsBooleanAlgebra( C ) ( property )

Returns: true or false

The property of C being a Boolean algebra.

#### 6.2 Operations

##### 6.2-1 MorphismFromDoubleNegation
 ‣ MorphismFromDoubleNegation( a ) ( attribute )

Returns: a morphism in $$\mathrm{Hom}(\neg\neg a, a)$$.

The argument is an object $$a$$. The output is the inverse $$\neg\neg a \rightarrow a$$ of the morphism to the double negation.

##### 6.2-2 MorphismFromDoubleNegationWithGivenDoubleNegation
 ‣ MorphismFromDoubleNegationWithGivenDoubleNegation( a, s ) ( operation )

Returns: a morphism in $$\mathrm{Hom}(\neg\neg a, a)$$.

The argument is an object $$a$$, and an object $$s = \neg\neg a$$. The output is the inverse $$\neg\neg a \rightarrow a$$ of the morphism to the double negation.

##### 6.2-3 MorphismToDoubleConegation
 ‣ MorphismToDoubleConegation( a ) ( attribute )

Returns: a morphism in $$\mathrm{Hom}(a, \ulcorner\ulcorner a)$$.

The argument is an object $$a$$. The output is the inverse $$a \rightarrow \ulcorner\ulcorner a$$ of the morphism from the double conegation.

##### 6.2-4 MorphismToDoubleConegationWithGivenDoubleConegation
 ‣ MorphismToDoubleConegationWithGivenDoubleConegation( a, s ) ( operation )

Returns: a morphism in $$\mathrm{Hom}(a, \ulcorner\ulcorner a)$$.

The argument is an object $$a$$, and an object $$r = \ulcorner\ulcorner a$$. The output is the inverse $$a \rightarrow \ulcorner\ulcorner a$$ of the morphism from the double conegation.

 ‣ AddMorphismFromDoubleNegation( C, F ) ( operation )

Returns: nothing

The arguments are a category $$C$$ and a function $$F$$. This operation adds the given function $$F$$ to the category for the basic operation MorphismFromDoubleNegation. $$F: ( a ) \mapsto \mathtt{MorphismFromDoubleNegation}(a)$$.

 ‣ AddMorphismFromDoubleNegationWithGivenDoubleNegation( C, F ) ( operation )

Returns: nothing

The arguments are a category $$C$$ and a function $$F$$. This operation adds the given function $$F$$ to the category for the basic operation MorphismFromDoubleNegationWithGivenDoubleNegation. $$F: ( a, s ) \mapsto \mathtt{MorphismFromDoubleNegationWithGivenDoubleNegation}(a, s)$$.

 ‣ AddMorphismToDoubleConegation( C, F ) ( operation )

Returns: nothing

The arguments are a category $$C$$ and a function $$F$$. This operation adds the given function $$F$$ to the category for the basic operation MorphismToDoubleConegation. $$F: ( a ) \mapsto \mathtt{MorphismToDoubleConegation}(a)$$.

 ‣ AddMorphismToDoubleConegationWithGivenDoubleConegation( C, F ) ( operation )

Returns: nothing

The arguments are a category $$C$$ and a function $$F$$. This operation adds the given function $$F$$ to the category for the basic operation MorphismToDoubleConegationWithGivenDoubleConegation. $$F: ( a, r ) \mapsto \mathtt{MorphismToDoubleConegationWithGivenDoubleConegation}(a, r)$$.

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