# Induced Semigroup Structures via (left) Translation

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 $$u\mapsto\zeta\circ u$$ 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.

• 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.. – Berci Nov 21 '18 at 0:34