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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).

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A completely regular semigroup is a semigroup where every element belongs to a subgroup of that semigroup.
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.

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