Integers under Subtraction do not form Semigroup

Theorem

Let $\struct {\Z, -}$ denote the algebraic structure formed by the set of integers under the operation of subtraction.

Then $\struct {\Z, -}$ is not a semigroup.

Proof

It is to be demonstrated that $\struct {\Z, -}$ does not satisfy the semigroup axioms.

We then have Subtraction on Numbers is Not Associative.

So, for example:

$3 - \paren {2 - 1} = 2 \ne \paren {3 - 2} - 1 = 0$

Thus it has been demonstrated that $\struct {\Z, -}$ does not satisfy Semigroup Axiom $\text S 1$: Associativity.

Hence the result.

$\blacksquare$

Sources

• 1970: B. Hartley and T.O. Hawkes: Rings, Modules and Linear Algebra ... (previous) ... (next): Chapter $1$: Rings - Definitions and Examples: Exercise $1$