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.

  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.