Set Elements

• Set ($𝕊$): A unordered collection of elements: $𝕊=\{a,b,c,\dots \}$
• Set Operations $\text{(+,⋅)}$: We consider two binary operations on $𝕊$: addition $\text{(+)}$ and multiplication $\text{(⋅)}$.
• Additive Identity ($0_{\text{(+)}}$ often written $0$): An element $i\in 𝕊$ is an additive identity if for all $a\in 𝕊$, $i+a=a=a+i$.
• Multiplicative Identity ($1_{\text{(⋅)}}$ often written $1$): An element $i\in 𝕊$ is a multiplicative identity if for all $a\in 𝕊$, $i\cdot a=a=a\cdot i$
• Additive Inverse ($a_{\text{(+)}}^{-1}$): For each $a\in 𝕊$, its additive inverse $a_{\text{(+)}}^{-1}$, often written $-a$, is defined as an element such that $a+a_{\text{(+)}}^{-1}=0_{\text{(+)}}=a_{\text{(+)}}^{-1}+a$ (i.e., additive identity)
• Multiplicative Inverse ($a_{\text{(⋅)}}^{-1}$): For each $a\in 𝕊$ that is invertible with respect to $\text{(⋅)}$, its multiplicative inverse $a_{\text{(⋅)}}^{-1}$, often written $a^{-1}$, is defined as an element such that $a\cdot a_{\text{(⋅)}}^{-1}=1_{\text{(⋅)}}=a_{\text{(⋅)}}^{-1}\cdot a$ (i.e., multiplicative identity)

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Element Operation Features

• Closed: A set $𝕊$ is closed under the $\text{(+)}$ operation if for every $a,b\in 𝕊$, it is the case that $a+b\in 𝕊$. Likewise, a set $𝕊$ is closed under the $\text{(⋅)}$ operation if for every $a,b\in 𝕊$, it is the case that $a\cdot b\in 𝕊$.
• Associative: For any $a,b,c\in 𝕊$, $(a+b)+c=a+(b+c)$
• Commutative: For any $a,b\in 𝕊$, $a+b=b+a$
• Distributive: If both $\text{(+)}$ and $\text{(⋅)}$ are defined (e.g. in a ring), then $a\cdot (b+c)=(a\cdot b)+(a\cdot c)$, and $(a+b)\cdot c=a\cdot c+b\cdot c$.

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Group Types

• Semigroup: A semigroup is a set of elements which is closed and associative on a single operation ($+$ or $\cdot$)

• Monoid: A monoid is a semigroup with an identity element $e$ (a neutral element that leaves any other element unchanged under the operation).

  (e.g., $0$ is the identity element for $+$ operator, $1$ is the identity element for the $\cdot$ operator)

• Group: A group is a monoid, and every element has an inverse with respect to the operation.
• Abelian Group: An abelian group is a group, plus its operation is commutative.