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Suppose we take an initial segment $X$ of the class of ordinals such that $\sup X$ is a limit ordinal (for example $\mathbb{N}$). I know there are several ways of making $X$ into a monoid, like extending canonically the definition of sum, product or maximum, for natural numbers (whenever those operations are closed in $X$). I never systematically studied monoids so maybe what I'm about to ask trivial but I was wondering the following:

1) How many (up to isomorphism) monoid structure we can give to $X$ ?

2) Does this depends on $X$ itself? Probably yes, as, for example, on the Naturals one could define as a binary operation "take the greatest common divisor", however I don't know if this would makes sense in any proper superset of the Naturals.

3) What if we restrain ourselves to the commutative case?

4) What if we restrain even more by asking for order preserving operations?

Any hint or reference is welcomed :)

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For $\Bbb{N}$, the answer to your four questions is the cardinality of the continuum. Indeed consider, for each subset $S$ of $\Bbb{N} - \{1\}$, the commutative monoid $M_S = (\Bbb{N}, *_S)$ defined by setting \begin{align} 1 *_S x &= x *_S 1 = x \text{ for all $x \in \Bbb{N}$} \\ x *_S y &= \begin{cases} x &\text{if $x = y$ and $x \in S$} \\ 0 &\text{otherwise} \end{cases} \end{align} Then the map $S \to M_S$ is clearly injective and thus one gets \begin{align} \mathfrak c =\text{Card}(\mathcal{P}(\Bbb{N}- \{1\})) &\leqslant \text{Card}\{ \text{Commutative monoids defined on $\Bbb{N}$}\} \\ &\leqslant \text{Card}\{ \text{Monoids defined on $\Bbb{N}$}\} \\ &\leqslant \text{Card}(\Bbb{N}^{\Bbb{N} \times \Bbb{N}}) = \mathfrak c \end{align} The ordered case does not change much, since if $M$ is a monoid, then $(M, =)$ is an ordered monoid. Thus \begin{align} \mathfrak c &= \text{Card}\{ \text{Monoids defined on $\Bbb{N}$}\} \\ &\leqslant \text{Card}\{ \text{Ordered monoids defined on $\Bbb{N}$}\} \\ &\leqslant \text{Card}\{ \text{Monoids defined on $\Bbb{N}$}\} \times \text{Card}\{ \text{Orders on $\Bbb{N}$}\} \leqslant \mathfrak c \end{align}

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