A internal binary operation on a set \(S\) is a [provisional cross-reference: def-function]
\begin{equation*}
-\cdot-:S\times S\to S,\text{given by } (x,y)\mapsto x\cdot y.
\end{equation*}
Convention7.2.
Let \(S\) be a set, and let \(* : S \times S \to S\) be a binary operation. If \(a, b\in S\text{,}\) then it would be proper to denote the result of doing the operation \(*\) to the pair \((a, b)\) by writing \(*((a, b))\text{.}\) Such notation is quite cumbersome, however, and would not look like familiar binary operations such as addition of numbers. Hence, we will write \(a * b\) instead of \(*((a, b))\text{.}\)
Remark7.3.
Internal binary operations are usually reffered to as “binary operations” as they are the most common form of binary operation. 1
Definition7.4.
A binary operation is closed
Example7.5.
Addition.
The function \(+:S\to S\) defined by \(+(a,b)=a+b\) is a binary operation for \(S\in\{\N,\Z,\Q,\R,\C\}\text{.}\)
Multiplication.
The function \(\cdot:S\to S\) defined by \(+(a,b)=a\cdot b\) (often shortened to \(ab\)) is a binary operation for \(S\in\{\N,\Z,\Q,\R,\C\}\text{.}\)
Composition.
Let \(F\) denote the set of all functions from a set \(S\) to itself. The function \(\circ:F\to F\) defined by \(\circ(f,g)=g\circ f\) is a binary operation.
Definition7.6.
A binary operation on a set \(S\) is commutative if \(a*b=b*a\) for all \(a,b\in S\text{.}\)
Definition7.7.
A binary operation on a set \(S\) is associative if \((a*b)*c=a*(b*c)\) for all \(a,b,c\in S\text{.}\)
Definition7.8.
A binary operation on a set \(S\) is idempotent if \(a * a=a\) for all \(a\in S\text{.}\)
Definition7.9.
Let \(*\) be a binary operation on a set \(S\text{.}\)
An element \(\ell\) is called a left identity if \(\ell * a=a\) for all \(a\in S\text{.}\)
An element \(r\) is called a right identity if \(r * a=a\) for all \(a\in S\text{.}\)
An element \(e\) is called an identity if it is both a left and a right identity.
Theorem7.10.
Any binary operation has at most one identity.
Corollary7.11.
If an operation has both a left identity and a right identity, then these two elements are equal.
Definition7.12.
Let \(*\) be a binary operation on a set \(S\) with identity \(e\text{,}\) and let \(a\in S\text{.}\)
An element \(b\) is called a left inverse of \(a\) if \(b * a=e\text{.}\)
An element \(c\) is called a right inverse if \(a * b=e\text{.}\)
An element that is both a left inverse and right inverse of \(a\) is called an inverse of \(a\text{.}\)
Proposition7.13.
Let \(*\) be an associative binary operation on a set \(S\text{.}\)
Any element in \(S\) has at most one inverse.
If \(a\in S\) has a left inverse \(b\) and a right inverse \(c\text{,}\) then these \(b=c\text{.}\)
SubsectionPartial Binary Operations
Definition7.14.
A partial function from a set \(X\) to a set \(Y\) is a function from a subset \(A\) of \(X\) to \(Y\text{.}\) This is denoted \(f:X\rightharpoonup Y\text{.}\)
Definition7.15.
A partial binary operation on a set \(S\) is a partial function
\begin{equation*}
-*-:S\times S\to S,\text{given by } (x,y)\mapsto x * y.
\end{equation*}
Example7.16.Division.
SubsectionExternal Binary Operations
Definition7.17.
The “internal” piece subtly implies the existence of some nebulous external binary operation, which we'll get to later.
