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We are looking for some (finite and infinite) examples of non-monoid semigroups $S$ containing a non-trivial subgroup $H$ satisfying the following conditions:

(1) $S$ has more than one left identity;

(2) $1_H$ is a left identity of $S$;

(3) There are some $h\in H$ such that $sh=1_H$ for some $s\neq h^{-1}$ (in $S$), where $ h^{-1}$ denotes the inverse of $h$ in $H$.

Note. Such examples were not found among the semigroups of functions, matrices, and similar to them.

M.H.Hooshmand
  • 2,529

1 Answers1

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Consider the semigroup $(S,\circ)$ where $S=\mathbb{R}\setminus\{0\}$ and $x\circ y=|x|y$. It is obvious that $H=(0,+\infty)$ is a subgroup of $S$. Now

(1) $S$ contains the left identities $\pm 1$, and no right identity.

(2) $1_H=1$ is a left identity of $S$.

(3) Putting $s=\frac{-1}{h}$, for every $h\in H$, we have $s\circ h=1_H$ and $s\neq h^{-1}$.

Also, see https://math.stackexchange.com/questions/433546/is-a-semigroup-g-with-left-identity-and-right-inverses-a-group#:~:text=Semigroup%20with%20left%20unit%20

M.H.Hooshmand
  • 2,529