Regular semiring (in the sense of von Neumann)

Question

A semiring $R$ is called regular (in the sense of von Neumann) if for each $a\in R$, there exists $x, y\in R$ such that $a+axa=aya.$

Reference here

Further, from the above link, I encounter that the semiring $\langle \Bbb Z_+ \cup \{0\}, \max, \cdot \rangle$ of non-negative integers with usual multiplication and $a+b=\max\lbrace a, b \rbrace$, is a regular semiring in the above sense. But I think this will be possible only if we take $x=y=1$. Now my question is: is taking $x=y$ in the above definition a valid consideration or not?

Answer 1

In the natural numbers (zero included) it holds $a\le a^2$, so taking $x=y=1$ is perfectly right.