What is a typical and clarifying example of a set that satisfies associativity?

Question

I understand what associativity is, and of course simple structures from elementary and high school mathematics like the natural numbers over addition are associative.

However, I think it would be clarifying to have an example of a mathematical structure that has almost no structure besides satisfying associativity, and that is also very simple.

Answer 1

If $X$ is any set, we can look at $$ \mathrm{Fun}(X) = \{f \mid f:X \to X\}. $$ This set is associative under composition. If $f,g,h \in \mathrm{Fun}(X)$, then $(f \circ g) \circ h = f \circ (g \circ h)$ because $$ ((f \circ g) \circ h)(x) = (f \circ g)(h(x)) = f(g(h(x))) = f((g \circ h)(x)) = f \circ (g \circ h)(x). $$ More generally, if you know what a homomorphism is, then you can also form $$ \mathrm{End}(X) = \{f\mid f:X \to X \text{ and $f$ is a homomorphism}\}. $$

Answer 2

I hope the folllowing simple one is a typical one. Let $S = \{a, b, c\}$ and define the product $*$ as in the given table:

Then $S$ is a finite semigroup (which I think you are looking for). It satisfies associativity requirement: $$x * (y * z) = (x * y) * z$$ where $x,y$ and $z$ are arbitrary elements of set $S$. Note that $S$ is just closed under $*$ before having associativity.

Answer 3

Here are a couple of examples that involve very little structure.

The operation of catenation on digit strings from a fixed alphabet (numeric or otherwise).

The left forgetful operation defined by $a*b=b$ for any $a,b$ belonging to some fixed set.

(And, of course, the corresponding right forgetful operation.)