2.1: Binary Operations and Structures

Remark

For \(*\) to be a binary operation on \(S\text{,}\) \(a*b\) must be well-defined and in \(\mathbf{S}\) for each \(a,b\in S\text{.}\) For instance, we cannot define a binary operation on \(\mathbb{R}\) by

\begin{equation*} a*b=\text{ the greatest number less than \(a+b\)} \end{equation*}

since there is no such “greatest number.” Nor can we define a binary operation on \(\mathbb{Z}\) by \(a*b=\dfrac{ab}{2}\text{,}\) since for, say, \(a=b=1 \in \mathbb{Z}\text{,}\) \(\dfrac{ab}{2}=\dfrac{1}{2} \not\in \mathbb{Z}\text{.}\)

Not every binary operation is denoted by \(*\text{.}\) In fact, many already have common notations: for instance, \(+\) on \(\mathbb{Z}\) or \(\circ\) on the set of functions from \(\mathbb{R}\) to \(\mathbb{R}\text{.}\) We will assume these common notations represent the “usual” binary operations we know them to mean, unless otherwise noted.
Do not mix up the \(*\) that indicates a binary operation and the superscript \(^*\) that indicates that we are only considering the nonzero elements of a given set (e.g., \(\mathbb{R}^*\)). You should be able to tell which type of \(*\) we are using from context and placement. Also, make sure you correctly place these symbols!

Definition: Associative

A binary operation \(*\) on a set \(S\) is associative if \((a*b)*c=a*(b*c)\) for all \(a,b,c\in S\text{.}\)

Remark

When a binary operation is associative, we can omit parentheses when operating on set elements. For example, \(+\) is associative on \(\mathbb{Z}\text{,}\) so we can unambiguously write the (equal) expressions \(1+(2+3)\) and \((1+2)+3\) as \(1+2+3\text{.}\)

Definition: Commutative

A binary operation \(*\) on set \(S\) is commutative if

\begin{equation*} a*b=b*a \end{equation*}

for all \(a,b\in S\text{.}\) We say that specific elements \(a\) and \(b\) of \(S\) commute if \(a*b=b*a\text{.}\)

Definition: Identity Element

Let \(\langle S,*\rangle\) be a binary structure. An element \(e\) in \(S\) is an identity element of \(\langle S,*\rangle\) if \(x*e=e*x=x\) for all \(x\in S\text{.}\)

Note

Sometimes an identity element of \(\langle S,*\rangle\) is referred to as an identity element of \(S\) under \(*\), or, when \(*\) is clear from context, simply as an identity element of \(S\).

Note

Not every binary structure contains an identity element! (Ex: \(\langle \mathbb{Z},-\rangle\text{.}\)

A natural question to ask is if a binary structure can have more than one identity element? The answer is no!

Theorem \(\PageIndex{1}\)

A binary structure \(\langle S, *\rangle\) has at most one identity element. That is, identity elements in binary structures, when they exist, are unique.

Proof: Sssume that ee and ff are identity elements of \(S\). Then since ee is an identity element, \(e∗f=f\) and since \(f\) is an identity element, \(e∗f=e\). Thus, \(e=f\).

Definition: Two-sided Inverse

Let \(\langle S, *\rangle\) be a binary structure with identity element \(e\text{.}\) Then for \(a\in S\text{,}\) \(b\) is an (two-sided) inverse of \(a\) in \(\langle S,*\rangle\) if \(a*b=b*a=e\text{.}\)

Note

We can also refer to such an element \(b\) as an inverse for \(a\) in \(S\) under \(*\), or, when \(*\) is clear from context, simply as an inverse of \(a\).

Note

Not every element in a binary structure with an identity element has an inverse!
If a binary structure does not have an identity element, it doesn't even make sense to say an element in the structure does or does not have an inverse!

Theorem \(\PageIndex{2}\)

Let \(\langle S, *\rangle\) be a binary structure with an identity element, where \(*\) is associative. Let \(a\in S\text{.}\) If \(a\) has an inverse, then its inverse is unique.

Proof: Let \(e\) be the identity element of \(S\). Suppose aa has inverses \(b\) and \(c\). Then \(a∗b=e\) so, multiplying both sides of the equation by \(c\) on the left, we have \(c∗(a∗b)=c∗e=c\). But since \(∗\) is associative, we have \(c∗(a∗b)=(c∗a)∗b=e∗b=b\). But \(b=c\).Thus, \(a\)'s inverse is unique.

We end by discussing the idea of closure under a binary operation.

Definition: Closed

Let \(\langle S,*\rangle\) be a binary structure and let \(T \subseteq S\text{.}\) Then \(T\) is said to be closed under \(*\) if \(t_1 * t_2 \in T\) whenever \(t_1,t_2\in T\text{.}\)

Example \(\PageIndex{1}\)

Consider the binary structure \(\langle \mathbb{M}_2(\mathbb{R}), +\rangle\text{,}\) where \(+\) denotes matrix addition. Let

\[T=\{A\in M_2(\mathbb{R}): A \mbox{ is invertible}\}.\]

We claim that \(T\) is not closed under \(+\text{.}\) Indeed, if we denote the identity matrix in \(\mathbb{M}_2(\mathbb{R})\) by \(I_2\text{,}\) then we observe that \(I_2, -I_2\in T\text{,}\) but \(I_2+(-I_2)\not\in T\text{,}\) since the zero matrix isn't invertible.

Example \(\PageIndex{2}\)

Consider the binary structure \(\langle \mathbb{M}_2(\mathbb{R}), \cdot\rangle\text{,}\) where \(\cdot\) denotes matrix multiplication. Again, let \(T\) be the set of all invertible matrices in \(\mathbb{M}_2(\mathbb{R})\text{.}\) We claim that \(T\) is closed under \(\cdot\,\text{.}\) Indeed, let \(A,B\in T\text{.}\) Then \(A\) and \(B\) are invertible, so their determinants are nonzero. Thus, \(\det(AB)=(\det A)(\det B)\neq 0,\) so \(AB\) is invertible, and hence \(AB\in T\text{.}\)