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# polyadic semigroup

Recall that a semigroup is a non-empty set, together with an associative binary operation on it. *Polyadic semigroups* are generalizations of semigroups, in that the associative binary operation is replaced by an associative $n$-ary operation. More precisely, we have

Definition. Let $n$ be a positive integer at least $2$. A *$n$-semigroup* is a non-empty set $S$, together with an $n$-ary operation $f$ on $S$, such that $f$ is associative:

$f(f(a_{1},\ldots,a_{n}),a_{{n+1}},\ldots,a_{{2n-1}})=f(a_{1},\ldots,f(a_{i},% \ldots,a_{{i+n-1}}),\ldots,f_{{2n-1}})$ |

for every $i\in\{1,\ldots,n\}$. A *polyadic semigroup* is an $n$-semigroup for some $n$.

An $n$-semigroup $S$ (with the associated $n$-ary operation $f$) is said to be *commutative* if $f$ is commutative. An element $e\in S$ is said to be an *identity element*, or an *$f$-identity*, if

$f(a,e,\ldots,e)=f(e,a,\ldots,e)=\cdots=f(e,e,\ldots,a)=a$ |

for all $a\in S$. If $S$ is commutative, then $e$ is an identity in $S$ if $f(a,e,\ldots,e)=a$.

Every semigroup $S$ has an $n$-semigroup structure: define $f:S^{n}\to S$ by

$f(a_{1},a_{n}\ldots,a_{n})=a_{1}\cdot a_{2}\cdots\cdot a_{n}$ | (1) |

The associativity of $f$ is induced from the associativity of $\cdot$.

Definition. An $n$-semigroup $S$ is called an *$n$-group* if, in the equation

$f(x_{1},\ldots,x_{n})=a,$ | (2) |

any $n-1$ of the $n$ variables $x_{i}$ are replaced by elements of $G$, then the equation with the remaining one variable has *at least one* solution in that variable. A *polyadic group* is just an $n$-group for some integer $n$.

$n$-groups are generalizations of groups. Indeed, a $2$-group is just a group.

###### Proof.

Let $G$ be a $2$-group. For $a,b\in G$, we write $ab$ instead of $f(a,b)$. Given $a\in G$, there are $e_{1},e_{2}\in G$ such that $ae_{1}=a$ and $e_{2}a=a$. In addition, there are $x,y\in G$ such that $xa=e_{2}$ and $ay=e_{1}$. So $e_{2}=xa=x(ae_{1})=(xa)e_{1}=e_{2}e_{1}=e_{2}(ay)=(e_{2}a)y=ay=e_{1}$.

Next, suppose $ae_{1}=ae_{3}=a$. Then the equation $e_{2}a=a$ from the previous paragraph as well as the subsequent discussion shows that $e_{1}=e_{2}=e_{3}$. This means that, for every $a\in G$, there is a unique $e_{a}\in G$ such that $e_{a}a=ae_{a}=a$. Since $e_{a}^{2}a=e_{a}(e_{a}a)=e_{a}a=a=ae_{a}=(ae_{a})e_{a}=ae_{a}^{2}$, we see that $e_{a}$ is idempotent: $e_{a}^{2}=e_{a}$.

Now, pick any $b\in G$. Then there is $c\in G$ such that $b=ce_{a}$. So $be_{a}=(ce_{a})e_{a}=ce_{a}^{2}=ce_{a}=b$. From the last two paragraphs, we see that $e_{a}=e_{b}$. This shows that there is a $e\in G$ such that $ae=ea=a$ for all $a\in G$. In other words, $e$ is the identity with respect to the binary operation $f$.

Finally, given $a\in G$, there are $b,c\in G$ such that $ab=ca=e$. Then $c=ce=c(ab)=(ca)b=eb=b$. In addition, if $ab_{1}=ab_{2}=e$, then, from the equation $ca=e$, we get $b_{1}=c=b_{2}$. This shows $b$ is the unique inverse of $a$ with respect to binary operation $f$. Hence, $G$ is a group. ∎

Every group has a structure of an $n$-group, where the $n$-ary operation $f$ on $G$ is defined by the equation (1) above. Interestingly, Post has proved that, for every $n$-group $G$, there is a group $H$, and an injective function $\phi:G\to H$ with the following properties:

1. $\phi(G)$ generates $H$

2. $\phi(f(a_{1},\ldots,a_{n}))=\phi(a_{1})\cdots\phi(a_{n})$

If we call the group $H$ with the two above properties a *covering group* of $G$, then Post’s theorem states that every $n$-group has a covering group.

From Post’s result, one has the following corollary: an $n$-semigroup $G$ is an $n$-group iff equation (2) above has *exactly one* solution in the remaining variable, when $n-1$ of the $n$ variables are replaced by elements of $G$.

# References

- HB R. H. Bruck, A Survey of Binary Systems, Springer-Verlag, 1966
- EP E. L. Post, Polyadic groups, Trans. Amer. Math. Soc., 48, 208-350, 1940, MR 2, 128
- WD W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z. 29, 1-19, 1928

## Mathematics Subject Classification

20N15*no label found*20M99

*no label found*

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