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N. Bourbaki formally defines the free magma $M(X)$ over a set $X$. However, it does not define the free unital magma over $X$, which I am denoting by $M^{\ast}(X)$ (maybe you know some more common notation...). I want to use this last notion to define free unital nonassociative algebras. It seems to me that it suffices to "add a unit" to the free magma. So, we would have $M^{\ast}(X)=M(X)\cup\{1\}$. I think that $M(X)\cup\{1\}$ satisfies the universal property of a free object. What do you think? Do you have any reference about this construction? Thanks in advance.

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    $\begingroup$ Is this not just the same as adding an identity to a free semigroup to get a free monoid? If so, this is trivial. $\endgroup$ – Carl-Fredrik Nyberg Brodda Feb 26 at 13:22
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    $\begingroup$ This seems to work fine. Just use the universal property to extend to M(X) and map 1 to 1 $\endgroup$ – Benjamin Steinberg Feb 26 at 15:13

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