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For order $2$, I have found 5. There are 16 maps from $\{a, b\} \times \{a, b\} \to\{a, b\}$. They form $10$ equivalence classes of non-isomorphic binary operations, $5$ of which are associative. Of these $2$ are monoids and $1$ is a group.They are

  1. The group $ \mathrm Z/2 $
  2. $\{ 0, 1\}$ with standard multiplication.
  3. $\{a, b\}$ with $x \circ y = a$ for all $x$ and $y$ $=a$ or $b$
  4. $\{a, b\}$ with $x \circ y = x$ for all $x$ and $y$ $=a$ or $b$
  5. $\{a, b\}$ with $x \circ y = y$ for all $x$ and $y$ $=a$ or $b$

Is $5$ the correct number of semigroups for $n=2$ ?

How many semigroups are there for $n=3$ ? No proof is needed, but a reference would be appreciated.

Stephen Meskin
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2 Answers2

1

Up to isomorphism and anti-isomorphism, there are 4 semigroups of order 2, and 18 of order 3. (An anti-isomorphism is essentially transposing the multiplication table, up to re-labeling; formally, an anti-isomorphism of semigroups is a bijection $\phi: S\rightarrow T$ where $\phi(s_1s_2)=\phi(s_2)\phi(s_1)$.)

Indeed, up to isomorphism and anti-isomorphism: $$ \begin{array}{c|r} \text{Size}&\text{Number of semigroups}\\ \hline 1& 1\\ 2& 4\\ 3& 18\\ 4& 126\\ 5& 1,160\\ 6& 15,973\\ 7& 836,021\\ 8& 1,843,120,128\\ 9& 52,989,400,714,478 \end{array} $$ For a reference, you can find this table in the documentation for the SmallSemi GAP package. (Like the Small Groups package, but for semigroups.)

user1729
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  • In 1) i'm using addition mod 2 so the core of the Cayley Table looks like $\bigl( \begin{smallmatrix} 0 & 1\1 & 0\end{smallmatrix}\bigr)$ . While in 2) I'm using multiplication so the corresponding table is $\bigl( \begin{smallmatrix} 0 & 0\0 & 1\end{smallmatrix}\bigr)$ – Stephen Meskin Dec 07 '17 at 12:23
  • @bof $0$ is the identity and is its own inverse. – Stephen Meskin Dec 07 '17 at 12:25
  • is not a group $1$ is the identity and here $0$ has no inverse.
    • – Stephen Meskin Dec 07 '17 at 12:32
  • Okay, I understand. The issue was that the table I found is up to isomorphism and "anti-isomorphism" (this took me a while to work out!). I've edited this into my answer. – user1729 Dec 07 '17 at 13:20
  • @StephenMeskin Sorry, by (1) I meant (2). – bof Dec 07 '17 at 23:01
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    @Zach Fixed. Thanks. – user1729 Dec 16 '17 at 22:21