2.1 Binary Operations
Let \(M\) be a set. A binary operation on \(M\) is a function
often written \((x,y)\mapsto x\cdot y\). A pair \((M,\cdot )\) consisting of a set \(M\) and a binary operation \(\cdot \) on \(M\) is called a magma.
Let \(M=\mathbb {Z}\) and let \(+:\mathbb {Z}\times \mathbb {Z}\to \mathbb {Z}\) be the function \((x,y)\mapsto x+y\). Then, \(+\) is a binary operation and, consequently, \((\mathbb {Z},+)\) is a magma.
Let \(n\) be an integer and set \(\mathbb {Z}_{\geq n}:=\{ x\in \mathbb {Z}\, |\, x\geq n\} \). Now suppose \(n\geq 0\). Then, for \(x,y\in \mathbb {Z}_{\geq n}\), \(x+y\in \mathbb {Z}_{\geq n}\). Consequently, \(\mathbb {Z}_{\geq n}\) with the operation \((x,y)\mapsto x+y\) is a magma. In particular, \(\mathbb {Z}_+\) is a magma under addition.
Let \(S=\{ 0,1\} \). There are \(16=4^2\) possible binary operations \(m:S\times S\to S\). Therefore, there are 16 possible magmas of the form \((S,m)\).
Let \(n\) be a non-negative integer and let \(\cdot :\mathbb {Z}_{\geq n}\times \mathbb {Z}_{\geq n}\to \mathbb {Z}_{\geq n}\) be the operation \((x,y)\mapsto xy\). Then \(\mathbb {Z}_{\geq n}\) is a magma. Similarly, the pair \((\mathbb {Z},\cdot )\) is a magma (where \(\cdot :\mathbb {Z}\times \mathbb {Z}\to \mathbb {Z}\) is given by \((x,y)\mapsto xy\)).
Let \(M_2(\mathbb {R})\) denote the set of \(2\times 2\) matrices with real entries. If
are two matrices, define
Then \((M_2({\mathbb {R}}), \circ )\) is a magma. The operation \(\circ \) is called matrix multiplication.
If \((M,\cdot )\) is a magma, then \(M\) is called the underlying set and \(\cdot \) is called the binary operation or sometimes the multiplication.
There is a substantial amount of abuse of notation that goes along with binary operations. For example, suppose \((M,\cdot )\) is a magma and \(m, n\in M\). Instead of writing \(m\cdot n\) we often omit the \(\cdot \) from the notation and write \(mn\) as in Example 2.5. Moreover, when referring to a magma \((M,\cdot )\), we often simply refer to the underlying set \(M\) and write the binary operation as \((x,y)\mapsto xy\). That way we avoid having to write down a name for the binary operation. So, for example, we say, “let \(M\) be a magma" when we should really say, “let \((M,\cdot )\) be a magma.” We use this abuse of notation in the following definition.
Let \(M\) be a magma. We say that \(M\) is commutative if, for all \(x,y\in M\), \(xy=yx\). We say that \(M\) is associative if, for all \(x,y,z\in M\), \((xy)z=x(yz)\). An element \(e\in S\) is an identity element if, for all \(m\in M\), \(em=me=m\).
There is another important product on \(M_2(\mathbb {R})\) called the Lie bracket. It is given by \((A,B)\mapsto [A,B]:=A\circ B-B\circ A\). It is not associative. To see this, set
Then
but
We write \(\mathfrak {gl}_2(\mathbb {R})\) for the magma consisting of the set \(M_2(\mathbb {R})\) equipped with the Lie bracket binary operation.
If \(M\) is a commutative magma, then sometimes we write the binary operation as \((m,n)\mapsto m+n\). We never use the symbol “\(+\)” for a binary operation which is not commutative. Also, if the binary operation is written “\(+\),” we never omit it from the notation. For example, while we write \(3\times 5\) as \((3)(5)\), we never write \(3+5\) as \((3)(5)\).
Let \(M\) be a magma. Then there is at most one identity element \(e\in S\).
Suppose \(e,f\) are identity elements. Then \(e=ef=f\).
If \(M\) is a commutative magma with binary operation \(+\) then it is traditional to let the symbol “\(0\)” denote the identity element. Otherwise, it is traditional to use the symbol “\(e\)” or the symbol “\(1\).”