Some More Properties of a Binary Operation

Beyond closure, a binary operation $*$ on a set $S$ may or may not satisfy four further properties.

Commutative property. $*$ is commutative if $a*b=b*a$ for all $a,b\in S$.

Associative property. $*$ is associative if $(a*b)*c=a*(b*c)$ for all $a,b,c\in S$.

Existence of identity property. An element $e\in S$ is the identity element of $S$ under $*$ if $a*e=a$ and $e*a=a$ for every $a\in S$.

Existence of inverse property. If an identity $e$ exists, and if for a given $a\in S$ there is a $b\in S$ with $a*b=e$ and $b*a=e$, then $b$ is the inverse element of $a$, written $b=a^{-1}$. (Note: $a^{-1}$ denotes this element abstractly -- it should never be read as the fraction $\tfrac1a$ unless the operation happens to be ordinary multiplication.)

Proving "for every" is hard; disproving it is easy. A property phrased with "for every"/"for all" must, strictly, be checked for every pair or triple in $S$ -- rarely practical by hand. But its negation only needs one counterexample: "there exists a pair/triple for which the property fails" is enough to disprove it. This is exactly how non-commutativity or non-associativity gets shown -- one specific numerical example that breaks the equality.

Uniqueness, proved in general.

Theorem 12.1 (Uniqueness of Identity). In an algebraic structure, an identity element, if it exists, is unique. Proof. Suppose $e_1,e_2$ are both identities for $(S,*)$. Treating $e_1$ as the identity acting on the element $e_2$: $e_2*e_1=e_1*e_2=e_2$. Treating $e_2$ as the identity acting on $e_1$: $e_1*e_2=e_2*e_1=e_1$. Comparing the two, $e_1=e_2$. $\blacksquare$

Theorem 12.2 (Uniqueness of Inverse). In an algebraic structure, the inverse of an element, if it exists, is unique. Proof. Let $a\in S$ have two inverses $a_1,a_2$, with identity $e$. Then $a_1=a_1*e=a_1*(a*a_2)=(a_1*a)*a_2=e*a_2=a_2$ (using associativity, then $a_1*a=e$, then $e*a_2=a_2$). So $a_1=a_2$. $\blacksquare$

Five fully worked patterns from the book, to internalise the checking method:

1. $+$ on $\mathbb Z$: closed, commutative, associative, identity $0$, inverse of $m$ is $-m$ -- every property holds.
2. $-$ on $\mathbb Z$: binary, but not commutative ($4-5\ne5-4$) and not associative ($(4-5)-7=-8\ne4-(5-7)=6$); no identity, hence no inverse.
3. $+$ on the even integers $\mathbb E$: every property holds exactly as for $+$ on $\mathbb Z$ (sum of two evens is even, identity $0$, inverse of $2k$ is $-2k$).
4. $+$ on the odd integers $\mathbb O$: not even binary -- the sum of two odd integers $(2m+1)+(2n+1)=2(m+n+1)$ is always even, so it leaves $\mathbb O$. …