2
votes
0answers
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 ...
2
votes
1answer
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 ...
0
votes
1answer
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 ...
4
votes
1answer
125 views

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 $u\in M$. ...
2
votes
2answers
193 views

Commutative monoid, unsure of how to deal with negative elements, inverses and subtraction

We are working with a commutative monoid. Subtraction might be useful for us. However, we're not sure how to proceed -- negative elements have no meaning. How do we deal with allowing subtraction ...