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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.

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Since the map $u\mapsto\zeta\circ u$ is an injection with range $M_\zeta$, we actually get a bijection $M\to M_\zeta$, and you just pulled over the semigroup operation of $M$.
Note that we could have done it with any other bijection $\phi:M\to N$, by defining $a\circ_N b:=\phi^{-1}(a)\circ_M\phi^{-1}(b)$.
Consequently, as semigroups, $M$ and $M_\zeta$ are isomorphic.

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  • $\begingroup$ Nevertheless, I've seen very similar and useful (!) ideas in e.g. functional analysis, defining a new inner product on the range of certain operators.. $\endgroup$ – Berci Nov 21 '18 at 0:34

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