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 identity)?
2026-03-27 18:05:47.1774634747
Monoid with inversion
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Yes, it is called a monoid with involution, though the phrase semigroup with involution is more popular. Involutions don't have to have anything to do with inverses.
If you wanted $a \cdot a^{-1} \cdot a = a$ and $a^{-1} \cdot a \cdot a^{-1} = a^{-1}$, then you get an inverse monoid.
For instance in the semigroup with 0 of all (possibly rectangular) matrices, the transpose is an involution, but the pseudo-inverse $A \mapsto A^t(AA^t)^{-1}$ makes it an inverse monoid once you fix several problems with my abridged definition).