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 -ary operation. More precisely, we have
Definition. Let be a positive integer at least . A -semigroup is a non-empty set , together with an -ary operation on , such that is associative:
for every . A polyadic semigroup is an -semigroup for some .
for all . If is commutative, then is an identity in if .
Every semigroup has an -semigroup structure: define by
The associativity of is induced from the associativity of .
Definition. An -semigroup is called an -group if, in the equation
any of the variables are replaced by elements of , then the equation with the remaining one variable has at least one solution in that variable. A polyadic group is just an -group for some integer .
-groups are generalizations of groups. Indeed, a -group is just a group.
Let be a -group. For , we write instead of . Given , there are such that and . In addition, there are such that and . So .
Next, suppose . Then the equation from the previous paragraph as well as the subsequent discussion shows that . This means that, for every , there is a unique such that . Since , we see that is idempotent: .
Now, pick any . Then there is such that . So . From the last two paragraphs, we see that . This shows that there is a such that for all . In other words, is the identity with respect to the binary operation .
Finally, given , there are such that . Then . In addition, if , then, from the equation , we get . This shows is the unique inverse of with respect to binary operation . Hence, is a group. ∎
Every group has a structure of an -group, where the -ary operation on is defined by the equation (1) above. Interestingly, Post has proved that, for every -group , there is a group , and an injective function with the following properties:
If we call the group with the two above properties a covering group of , then Post’s theorem states that every -group has a covering group.
From Post’s result, one has the following corollary: an -semigroup is an -group iff equation (2) above has exactly one solution in the remaining variable, when of the variables are replaced by elements of .
- 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
|Date of creation||2013-03-22 18:37:47|
|Last modified on||2013-03-22 18:37:47|
|Last modified by||CWoo (3771)|