# How are bands completely regular?

In Howie's Fundamentals of Semigroup Theory, he states that any band $$B$$ is completely regular if we define $$x'=x$$, which gives us the identities $$(x')'=x$$, $$xx'x=x$$, and $$xx'=x'x$$. I don't understand what $$x'$$ is supposed to be, is it an inverse of $$x$$? And if it is, how does it make sense to just define what the inverse is, without considering the binary operation $$B$$ is equipped with?

See

Howie, John M., Fundamentals of semigroup theory, London Mathematical Society Monographs. New Series. 12. Oxford: Clarendon Press. x, 351 p. (1995). ZBL0835.20077. : p. 113, Section 4.4 Bands (paragraph before it, and start of the paragraph after the section starts).

Since bands are idempotent semigroups, i.e., a semigroup is a band iff it satisfies the equation $$x \cdot x \approx x,$$ it follows that, in rather trivial way, each element of the band is part of a subgroup of the band (semigroup): the sub-semigroup $$\langle \{x\}, \cdot \rangle$$ has the structure of group if we define $$x$$ to be the unit and the inverse to be the identity map. Indeed, this structure satisfies any identity that you can com up with in the language of semigroups.
So yes, $$x'$$ denotes the inverse of $$x$$ (only within that singular subgroup of the band), and the binary operation is the same as the one of the band.