Owen wrote:

> I would answer "\\(\mathbb{Z}/2\\)" to both Puzzle 102' and Puzzle 102''—but I would mean the group under addition in the first case, and the monoid under multiplication in the second!

Okay, good! If you want to be polite to mathematicians, you should say \\( (\mathbb{Z}/2, +) \\) as your answer to Puzzle 102\\({}^\prime\\) and
\\( (\mathbb{Z}/2, \cdot ) \\) as your answer to Puzzle 102\\({}^{\prime\prime}\\) - that's how mathematicians distinguish between the same set made into monoids with two different operations.

If you just write \\(\mathbb{Z}/2\\), all mathematicians will assume you mean the group \\( (\mathbb{Z}/2, +) \\).

Another nice name for the monoid \\( ( \mathbb{Z}/2, \cdot ) \\) is \\( (\textbf{Bool}, \wedge) \\), since we can think of 0 as "false" and 1 as "true", and then multiplication becomes "and".

This monoid is also isomorphic to \\( (\textbf{Bool}, \vee) \\), but now the identity element is "false", not "true"!

Another nice name for the group \\( (\mathbb{Z}/2, +) \\) is \\( (\textbf{Bool}, \mathrm{XOR}) \\), since ["exclusive or"](https://en.wikipedia.org/wiki/Exclusive_or) makes the Booleans into a group that's isomorphic to \\( (\mathbb{Z}/2, +) \\).

And finally, if you make \\(\mathbb{Z}/2\\) into a _ring_ with both addition and multiplication as its operations, mathematicians often call it \\(\mathbb{F}_2\\), because it's the [field with two elements](https://en.wikipedia.org/wiki/GF(2)). You can also think of this field as the Booleans with "exclusive or" as addition and "and" as multiplication.