Is left inverse implying right inverse in matrix a property of structure?

Question

If $A$ is a square matrix and there exists a square matrix $B$ such that $AB =1$, than it is known that $BA=1$. This property is proved with some properties from linear algebra. Although I've never seen it be proved just by structures of matrix multiplication, I couldn't find a counterexample of a set with structures of matrix multiplication but left inverse doesn't imply right inverse.

To be more specific, let $X$ be a set and binary operation $\cdot$ is defined on $X$. If $\cdot$ is associative and $X$ has left and right identity(which will be the same), than does $A \cdot B = 1$ for some $A, B\in X$ implies $B \cdot A = 1$?

If not, what other properties of matrix multiplication should we add to this structure of $(X,\cdot)$ in order to get the property?

Answer 1

Take $X = \{f:\mathbb R\to\mathbb R\}$ equipped with composition $\circ$. Can you think of a function that is surjective but not injective?

Answer 2

If you just consider one operation, that is, if you work with monoids, the answer is no. A counterexample is the bicyclic monoid, which is the quotient of the free monoid on two generators $a$ and $b$ under the relation $ab = 1$.

However, let me answer your last question:

What other properties of matrix multiplication should we add to this structure of $(X,\cdot)$ in order to get the property?

It turns out that the property still holds if you work on a commutative semiring, which is, roughly speaking, a ring without subtraction. For a proof, see [1].

[1] Reutenauer, Christophe; Straubing, Howard. Inversion of matrices over a commutative semiring. J. Algebra 88 (1984), no. 2, 350--360.