@JakeGillberg wrote:

> How about an "almost-category" satisfying everything but the left identity law?

Here is one based ideas from the web. Let the morphisms of our one-object category be the integers \\(\mathbb{Z}\\).

Define the binary operation by a;b = |a|*b.

It's easy to show that this is associative. (So, a semigroup.)

1 is a left identity. But there's no right identity.

> How about an "almost-category" satisfying everything but the left identity law?

Here is one based ideas from the web. Let the morphisms of our one-object category be the integers \\(\mathbb{Z}\\).

Define the binary operation by a;b = |a|*b.

It's easy to show that this is associative. (So, a semigroup.)

1 is a left identity. But there's no right identity.