# Tagged Questions

47 views

### Taking the (pseudo)inverse of a monoid operation.

Let $M$ be a monoid with binary operation $f : M \times M \to M$. I'm interested in functions $g : M \to M\times M$ that obey the property: $$f(g(m)) = m$$ I want to understand what all of the ...
64 views

### Monoid with inversion

Is there a name for monoid with operation $a\mapsto a^{-1}$ conforming the equations $(a^{-1})^{-1}=a$ and $(b\cdot a)^{-1} = a^{-1}\cdot b^{-1}$? (with no requirement that $a^{-1}\cdot a$ would be ...
248 views

### Identity element, invertible and inverse elements in $(\mathbb Z_7 \times \mathbb Z_{10}, \odot)$

Let $T=\mathbb Z_7 \times \mathbb Z_{10}$ and let $\odot$ be the operation defined as follows: \begin{aligned} (a,b)\odot(c,d) = (2+a+c, 3bd)\end{aligned} Find the identity element, the inverse ...
### If $z$ is the unique element of a monoid such that $uzu=u$, is $u$ invertible?
This question is a follow-up to this one. I tried to check whether the same statement as discussed for rings there is true for monoids too, but without success. Let $M$ be a monoid and $u\in M$. ...