Example of an associative binary operation, without identities or inverses.

Question

In essence, I am looking for an example of a semigroup or a semicategory (closure is not that important, but it is useful) that is NOT a monoid or category.

Hopefully, there is a neat and simple-to-understand example of an only-associative operation on an infinite set.

Edit: Also, non-commutative!

Edit #2: From all the answers, it seems there are no associative, non-identity, non-inverse operations on the reals such that for $x,y \in \mathbb{R}$ the operation is $x \cdot y = f(x,y)$ where $f(x,y)$ is a some analytically expressible function, such that the reals form a semi-group under that operation.

Answer 1

How about strictly upper triangular matrices with matrix multiplication? Matrix multiplication is clearly associative and the product of two strictly upper triangular matrices is again strictly upper triangular. There are no inverses since these matrices have nontrivial kernel and there is no identity since the identity matrix is not strictly upper triangular.

Answer 2

Consider the left-identity semigroup. It can be defined on a set $S$ as follows:

for $a,b\in S$ let $a\star b=b$. this algebraic structure is a semigroup but it has no identity (although every element is a left-identity).

Answer 3

Continuous functions with convolution product form an algebraic structure where the product is associative but has no identity.

Given any set $X$ you can give it the following operation $$* \colon X \times X \to X$$ where $x*y=y$ for every pair $x,y \in X$. It's easy to see that this operation is associative but it clearly doesn't have any identity.

Answer 4

The set of all square matrices of order $n$ over $2\Bbb Z$ under matrix multiplication is an infinite non commutative semigroup without left or right identities. The same if you consider, instead of $2\Bbb Z$, any other infinite ring without identity, commutative or not. Also, in $2\Bbb Z$ no element is invertible.