If $N$ is a subset of a monoid $(M,\bot ,e)$ with identity element $\epsilon$ with respect $\bot$ then does the equality $\epsilon=e$ holds?

Question

If $(M,\bot,e)$ is a monoid then it is usually to say that $N$ in $\mathcal P(M)$ is a submonoid of $(M,\bot, e)$ if it is closed under $\bot$ and it contains $e$. So by this definition I suspect that if $(N,\bot)$ is a subgroup with a identity element $\epsilon$ then the equality $$ \epsilon=e $$ cannot hold but unfortunately I did not find any counterexample about so that I thought to put a specific question. Could someone help me, please?

Answer 1

$ \newcommand\Z{\mathbb Z} $If there is $\epsilon \in M$ which is idempotent ($\epsilon^2 = \epsilon$) then $\{\epsilon\}$ is a monoid with identity $\epsilon$, but moreover so is $\epsilon M\epsilon$. This is particulary used when $M$ is a ring in which case its called a corner ring. I'm not sure if "corner monoid" is standard but let's go with that. Note that $\epsilon = 0$ gives us $0M0 = \{0\}$.

For a more concrete example consider the multiplicative monoid $M = \Z/20\Z$. Then $5^2 = 25 = 5$ is idempotent and $$ 5M5 = 5M = \{0, 5, 10, 15\} $$ is a monoid that has $5$ as identity.

For a noncommutative example, consider $3{\times}3$ matrices over any ring $R$. The matrix $$\epsilon = \begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0 \end{pmatrix}$$ is idempotent and so $$ \epsilon M\epsilon = \left\{\begin{pmatrix} a & b & 0 \\ c & d & 0 \\ 0 & 0 & 0\end{pmatrix} \;:\; a, b, c, d \in R\right\} $$ is a monoid with identity $\epsilon$.

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All subsets of $M$ which are monoids with different identity are submonoids of its corner monoids. This is because if $\epsilon \in M$ is the identity of some $N \subseteq M$ then $\epsilon^2 = \epsilon$ and by virtue of being the identity of $N$ we get $$ N = \epsilon N\epsilon \subseteq \epsilon M\epsilon. $$