Polyadic group

Polyadic group ( $n$ ${\ displaystyle n}$ $n$ -ary group) in general algebra is a generalization of the concept of a group using $n$ ${\ displaystyle n}$ $n$ -ary operation instead of binary .

Polyadic groups became an object of independent study, starting with the work of W. Dörnte. ^[one]

Example

An example of a ternary group with three elements a , b and c .

aaa=a,aab=b,aac=c,aba=c,abb=a,abc=b,aca=b,acb=c,acc=a,

{\ displaystyle aaa = a, aab = b, aac = c, aba = c, abb = a, abc = b, aca = b, acb = c, acc = a,}

aaa=a,aab=b,aac=c,aba=c,abb=a,abc=b,aca=b,acb=c,acc=a,

baa=b,bab=c,bac=a,bba=a,bbb=b,bbc=c,bca=c,bcb=a,bcc=b,

{\ displaystyle baa = b, bab = c, bac = a, bba = a, bbb = b, bbc = c, bca = c, bcb = a, bcc = b,}

baa=b,bab=c,bac=a,bba=a,bbb=b,bbc=c,bca=c,bcb=a,bcc=b,

caa=c,cab=a,cac=b,cba=b,cbb=c,cbc=a,cca=a,ccb=b,ccc=c.

{\ displaystyle caa = c, cab = a, cac = b, cba = b, cbb = c, cbc = a, cca = a, ccb = b, ccc = c.}

caa=c,cab=a,cac=b,cba=b,cbb=c,cbc=a,cca=a,ccb=b,ccc=c.

Literature

↑ Dörnte, W. Untersuchungen über einen verallgemeinerten Gruppenbegrieff // Math. Z .. - 1928. - Vol. 29. - P. 1-19.

[Bra-1] Dörnte, W. Untersuchungen über einen verallgemeinerten Gruppenbegrieff // Math. Z .. - 1928. - Vol. 29. - P. 1-19.

Polyadic group

Example

Literature

More articles: