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I have a semigroup $G$ and I want to construct a monoid $M$ that has every element in $G$ plus an identity. I want to call this the free monoid generated from $G$, but that means something completely different. Is there a standard name for this construction?

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    "Unitisation of $G$", perhaps?2012-11-12

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nLab calls this the unitalization.

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Usually from a semigroup $S$ you construct a monoid $S^1$ by adding an unity to $S.$ That is $S^1= S\cup {1},$ where the multiplication in $S^1$ is given by $x\cdot y=xy$ for $x,y\in S$ and $x\cdot 1=1\cdot x=x$ for all $x\in S.$

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    Okay, thanks for the help2012-11-12