polyadic semigroup

Recall that a semigroupPlanetmathPlanetmath is a non-empty set, together with an associative binary operationMathworldPlanetmath on it. Polyadic semigroups are generalizationsPlanetmathPlanetmath of semigroups, in that the associative binary operation is replaced by an associative n-ary operationMathworldPlanetmath. 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:


for every i{1,,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 commutativePlanetmathPlanetmathPlanetmath if f is commutative. An element eS is said to be an identity elementMathworldPlanetmath, or an f-identityPlanetmathPlanetmathPlanetmathPlanetmath, if


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

Every semigroup S has an n-semigroup structureMathworldPlanetmath: define f:SnS by

f(a1,an,an)=a1a2an (1)

The associativity of f is induced from the associativity of .

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

f(x1,,xn)=a, (2)

any n-1 of the n variablesMathworldPlanetmath xi 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.


Let G be a 2-group. For a,bG, we write ab instead of f(a,b). Given aG, there are e1,e2G such that ae1=a and e2a=a. In additionPlanetmathPlanetmath, there are x,yG such that xa=e2 and ay=e1. So e2=xa=x(ae1)=(xa)e1=e2e1=e2(ay)=(e2a)y=ay=e1.

Next, suppose ae1=ae3=a. Then the equation e2a=a from the previous paragraph as well as the subsequent discussion shows that e1=e2=e3. This means that, for every aG, there is a unique eaG such that eaa=aea=a. Since ea2a=ea(eaa)=eaa=a=aea=(aea)ea=aea2, we see that ea is idempotentMathworldPlanetmathPlanetmath: ea2=ea.

Now, pick any bG. Then there is cG such that b=cea. So bea=(cea)ea=cea2=cea=b. From the last two paragraphs, we see that ea=eb. This shows that there is a eG such that ae=ea=a for all aG. In other words, e is the identity with respect to the binary operation f.

Finally, given aG, there are b,cG such that ab=ca=e. Then c=ce=c(ab)=(ca)b=eb=b. In addition, if ab1=ab2=e, then, from the equation ca=e, we get b1=c=b2. This shows b is the unique inverseMathworldPlanetmathPlanetmathPlanetmathPlanetmath 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 ϕ:GH with the following properties:

  1. 1.

    ϕ(G) generates H

  2. 2.


If we call the group H with the two above properties a covering group of G, then Post’s theoremMathworldPlanetmath 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.


  • 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
Title polyadic semigroup
Canonical name PolyadicSemigroup
Date of creation 2013-03-22 18:37:47
Last modified on 2013-03-22 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 n-semigroup
Synonym n-group
Defines n-semigroup
Defines n-group
Defines polyadic group
Defines covering group