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Let $(M,circ)$ be any semigroup satisfying the following properties:

P-1: $text{For every } x,y,z in M text{, if } z circ x = z circ y , text{ then } , x = y$.

If $zeta in M$ we define $M_zeta = {m in M ; | ; m = zeta circ u}$.

We can define a binary operation $circ_zeta$ on $M_zeta$ as follows:

$tag 1 (zeta circ u) circ_zeta (zeta circ v) = zeta circ (u circ v)$

It is easy to see that $circ_zeta$ is an associative operation and that the semigroup $M_zeta$ also satisfies property $text{P-1}$.

I can think of several other properties of $M$ that would also hold true for $M_zeta$.

Is there any developed theory that analyzes these induced algebraic
structures?

Any answers that contain books/papers as well any results would be
helpful.

Since the map $umapstozetacirc u$ is an injection with range $M_zeta$, we actually get a bijection $Mto M_zeta$, and you just pulled over the semigroup operation of $M$.

Note that we could have done it with any other bijection $phi:Mto N$, by defining $acirc_N b:=phi^{-1}(a)circ_Mphi^{-1}(b)$.

Consequently, as semigroups, $M$ and $M_zeta$ are isomorphic.