# What should I call this commutative monoid of order three?

I'm looking for a name for the monoid given by the following table:

$$\begin{array}{c|ccc}&1&a&b\\ \hline 1&1&a&b\\ a&a&1&b\\ b&b&b&b \end{array}$$

Is there a name that would be understandable to an undergraduate student who hasn't read anything about semigroups but has had a first course in algebra and knows what a semigroup/monoid is? What name would it be good to go under in a list of order-three semigroups?

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I'd call it $\{-1,0,1\}$, but I don't know very much about abstract algebra... – Rahul Narain Jul 2 '12 at 21:54
@RahulNarain Right. Thank you. This is really embarrassing. – user23211 Jul 2 '12 at 21:56
Call it Diego, just for kicks. – Asaf Karagila Jul 2 '12 at 22:13
@Asaf I haven't been so embarrassed for quite some time. Mocking may lead to an attempt at burying my head in the floor. That's probably safer than using sand, but still not a good idea. :) – user23211 Jul 2 '12 at 22:18
@ymar, if you prefer we can give it a Polish name, Stanislaw, perhaps. – Asaf Karagila Jul 2 '12 at 22:26
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## 2 Answers

Call it : $(\mathbb{Z}/3\mathbb{Z},*)$

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Oh... Thanks! The way I looked at it I did not notice that $b$ was a zero element. – user23211 Jul 2 '12 at 21:55
it was the first think i saw :) – hassan Jul 2 '12 at 21:58

In computer science I would call it $\mathbb{Z}_2$ with errors, i.e. $\langle \{0,1,\bot\},+,0\rangle$. In math I would follow hassan's idea.

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Thanks. So if $G$ is a group, then $G^0$ ($G$ with zero adjoined) is called "$G$ with errors"? – user23211 Jul 2 '12 at 22:24

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@ymar Not zero, but an element that spreads like a virus (usually denoted by $\bot$). Any operation can result in an error $\bot$ (e.g. if you divide $1$ by $0$ in some algebra containing division) and if the "error result" is passed to any other expression, it propagates, i.e. the result of that expression has to be $\bot$ as well. Indeed, that kind of structures are denoted $G^\bot$, and called "$G$ with errors" or "$G$ with error handling". This is very similar to NaNs or monad `Maybe` from Haskell language (e.g. $G^\bot \cong \mathtt{Maybe}\ G$). – dtldarek Jul 3 '12 at 6:09
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