Back

〈a, b | aaaab=1, abbbbb=1〉

Monoid presentation of length 11

Properties

Isomorphic to ℤ₁₉
Isomorphic to commutative presentation: 〈a | 19a=0〉

Completion parameters

Reduction order: left-to-right a/b

Inverses of generators

a^-1 = a¹⁸
b^-1 = a⁴

Complete rewriting system

a¹⁹ ⇒ 1
b ⇒ a¹⁵

Cayley table

	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
1	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸
a	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1
a²	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a
a³	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²
a⁴	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³
a⁵	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴
a⁶	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵
a⁷	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶
a⁸	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷
a⁹	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸
a¹⁰	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹
a¹¹	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰
a¹²	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹
a¹³	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²
a¹⁴	a¹⁴	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³
a¹⁵	a¹⁵	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴
a¹⁶	a¹⁶	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵
a¹⁷	a¹⁷	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶
a¹⁸	a¹⁸	1	a	a²	a³	a⁴	a⁵	a⁶	a⁷	a⁸	a⁹	a¹⁰	a¹¹	a¹²	a¹³	a¹⁴	a¹⁵	a¹⁶	a¹⁷

Right Cayley graph

Others with same cardinality

5 unique, 9 total

Length:	Presentation:	Description:	Related:
10	〈a, b \| aaa=a, abbba=b〉	Finite non-commutative monoid with 19 elements
10	〈a, b \| aba=b, bab=aaa〉	Finite non-commutative monoid with 19 elements
11	〈a, b \| aaab=b, abbba=a〉	Finite non-commutative monoid with 19 elements	3 isomorphic
11	〈a, b \| aaa=ab, bbab=bb〉	Finite non-commutative monoid with 19 elements	1 isomorphic
11	〈a, b \| aaa=bb, aaba=ab〉	Finite non-commutative monoid with 19 elements

Other isomorphic instances

20 total

Length:	Presentation:
11	〈a, b \| aaaab=1, babbbb=1〉
11	〈a, b \| aaaab=1, bbabbb=1〉
11	〈a, b \| aaaab=1, bbbabb=1〉
11	〈a, b \| aaaab=1, bbbbab=1〉
11	〈a, b \| aaaab=1, bbbbba=1〉
11	〈a, b \| aaaba=1, abbbbb=1〉
11	〈a, b \| aaaba=1, babbbb=1〉
11	〈a, b \| aaaba=1, bbabbb=1〉
11	〈a, b \| aaaba=1, bbbabb=1〉
11	〈a, b \| aaaba=1, bbbbab=1〉
11	〈a, b \| aaaba=1, bbbbba=1〉
11	〈a, b \| aabaa=1, abbbbb=1〉
11	〈a, b \| aabaa=1, babbbb=1〉
11	〈a, b \| aabaa=1, bbabbb=1〉
11	〈a, b \| aaab=1, bbbbbb=a〉
11	〈a, b \| aaba=1, bbbbbb=a〉
11	〈a, b \| aaa=b, abbbbbb=1〉
11	〈a, b \| aaa=b, babbbbb=1〉
11	〈a, b \| aaa=b, bbabbbb=1〉
11	〈a, b \| aaa=b, bbbabbb=1〉