polyadic semigroup
Recall that a semigroup^{} is a nonempty 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 nonempty set $S$, together with an $n$ary operation $f$ on $S$, such that $f$ is associative:
$$f(f({a}_{1},\mathrm{\dots},{a}_{n}),{a}_{n+1},\mathrm{\dots},{a}_{2n1})=f({a}_{1},\mathrm{\dots},f({a}_{i},\mathrm{\dots},{a}_{i+n1}),\mathrm{\dots},{f}_{2n1})$$ 
for every $i\in \{1,\mathrm{\dots},n\}$. 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
$$f(a,e,\mathrm{\dots},e)=f(e,a,\mathrm{\dots},e)=\mathrm{\cdots}=f(e,e,\mathrm{\dots},a)=a$$ 
for all $a\in S$. If $S$ is commutative, then $e$ is an identity in $S$ if $f(a,e,\mathrm{\dots},e)=a$.
Every semigroup $S$ has an $n$semigroup structure^{}: define $f:{S}^{n}\to S$ by
$$f({a}_{1},{a}_{n}\mathrm{\dots},{a}_{n})={a}_{1}\cdot {a}_{2}\mathrm{\cdots}\cdot {a}_{n}$$  (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
$$f({x}_{1},\mathrm{\dots},{x}_{n})=a,$$  (2) 
any $n1$ 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 $a{e}_{1}=a$ and ${e}_{2}a=a$. In addition^{}, there are $x,y\in G$ such that $xa={e}_{2}$ and $ay={e}_{1}$. So ${e}_{2}=xa=x(a{e}_{1})=(xa){e}_{1}={e}_{2}{e}_{1}={e}_{2}(ay)=({e}_{2}a)y=ay={e}_{1}$.
Next, suppose $a{e}_{1}=a{e}_{3}=a$. 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 ${e}_{a}a=a{e}_{a}=a$. Since ${e}_{a}^{2}a={e}_{a}({e}_{a}a)={e}_{a}a=a=a{e}_{a}=(a{e}_{a}){e}_{a}=a{e}_{a}^{2}$, 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 $b=c{e}_{a}$. So $b{e}_{a}=(c{e}_{a}){e}_{a}=c{e}_{a}^{2}=c{e}_{a}=b$. 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 $a{b}_{1}=a{b}_{2}=e$, 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 $\varphi :G\to H$ with the following properties:

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

2.
$\varphi (f({a}_{1},\mathrm{\dots},{a}_{n}))=\varphi ({a}_{1})\mathrm{\cdots}\varphi ({a}_{n})$
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 $n1$ of the $n$ variables are replaced by elements of $G$.
References
 HB R. H. Bruck, A Survey of Binary Systems, SpringerVerlag, 1966
 EP E. L. Post, Polyadic groups, Trans. Amer. Math. Soc., 48, 208350, 1940, MR 2, 128
 WD W. Dörnte, Untersuchungen über einen verallgemeinerten Gruppenbegriff, Math. Z. 29, 119, 1928
Title  polyadic semigroup 

Canonical name  PolyadicSemigroup 
Date of creation  20130322 18:37:47 
Last modified on  20130322 18:37:47 
Owner  CWoo (3771) 
Last modified by  CWoo (3771) 
Numerical id  10 
Author  CWoo (3771) 
Entry type  Definition 
Classification  msc 20N15 
Classification  msc 20M99 
Synonym  nsemigroup 
Synonym  ngroup 
Defines  $n$semigroup 
Defines  $n$group 
Defines  polyadic group 
Defines  covering group 