Going forward the set $\{0,1\}$ will be referred to as “Boolean Logic”

Logical Operators

Here is a list of logical operators.

• $\wedge$: And
• $\vee$: Or
• $\otimes$: Xor
• $\Rightarrow$: Implies
• $⟺$: Biconditional

Basic Algebras Derived from Logical Operators

Magmas

$(\{0,1\},\vee )$

Boolean logic under the or operation

1. is a semigroup
2. is a monoid, and the identity is $0$
3. is not a group since $1$ has no inverse

$(\{0,1\},\wedge )$

Boolean logic under the and operation

1. Boolean and has an Isomorphism with Boolean or by way of the function that applies the “Not” unary operation to the input.
2. Identity element is $1$ as it is what $0$ maps to in the isomorphism

$(\{0,1\},\Rightarrow )$

Boolean logic under the implies operation

1. is not a semigroup :( since it is not associative

$(\{0,1\},\otimes )$

Boolean logic under the xor operation

1. Is a semigroup
2. is a monoid
3. is a group since all elements are invertable

$(\{0,1\},⟺)$

Boolean logic under the biconditional operation (also called iff for if and only if)

1. There exists an Isomorphism between boolean xor and boolean iff therefore they share the same properties

Rings/Double Magmas

We can combine different binary logic groups and semigroups to get ring candidates and analyze them. See also Boolean Rings.

$(\{0,1\},\otimes ,\vee )$

Boolean logic under xor, or

1. Is NOT a ring since not all elements distribute

$(\{0,1\},\otimes ,\wedge )$

Boolean logic under xor, and

1. Is a Ring!
   1. Distributive property holds for all elements
   2. $(\{0,1\},\otimes )$ is commutative
2. Is a Field!
   1. $(\{0,1\},\wedge )$ is commutative
   2. All elements of $(\{0,1\},\wedge )$ excluding the $\otimes$ identity, $0$, ($0\wedge 0\not\equiv 1$) have an inverse.
   3. We already showed $(\{0,1\},\wedge )$ has an identity!

$(\{0,1\},⟺,\vee )$

Boolean logic under iff, or

1. Is isomorphic to $(\{0,1\},\otimes ,\wedge )$ since
   1. $(\{0,1\},⟺)\cong (\{0,1\},\otimes )$
   2. $(\{0,1\},\vee )\cong (\{0,1\},\wedge )$
   3. And they have the same isomorphism, $f(x)=\neg x$

Variety Generation

Members of the variety generated by $\mathbb{B}=(\{0,1\},\otimes ,\wedge ,\neg ,0)$ We’ve included $\neg$ and $0$ because it is important to include ALL operations when talking about varieties.

Are called a Boolean Algebras See Classification of Boolean Algebras

Some important laws of the variety of Boolean Algebras are:

1. All the laws that apply to rings
2. AND is Idempotent, $A\wedge A=A$
3. AND’s identity is $\neg 0$
4. $A\otimes A=0$

Finite Algebras of $\mathbb{B}$

$\mathbb{B}\times \mathbb{B}$

$(\{(0,0),(0,1),(1,0),(1,1)\},\otimes ,\wedge ,\neg ,0)$

Homomorphic Images of $\mathbb{B}\times \mathbb{B}$

1. $(\{0,1\},⟺,\vee ,\neg ,1)$ is an isomorphism so its basically the same algebra
2. $\mathbb{B}$ using Projections
3. The Trivial Algebra ${\mathbb{B}}^0$ $(\{a\},\otimes ,\wedge ,\neg ,0)$ in otherwords $a=a$

Every finite boolean algebra is like this. There is no boolean algebra with 3 elements. Ternary logic is incompatible with Boolean algebra. (This is a separate non trivial theorem). You can do it using Legranges theorem to demonstrate why $\mathbb{B}$ is a subset of all algebras in this variety so it must have an even number of elements except the trivial one.

Infinite Algebras of $\mathbb{B}$

${\mathbb{B}}^{\mathcal{I}}$ where $\mathcal{I}$ in infinite

There may be countable subalgebras