〈a, b | aaa=a, bbbbbb=a〉

Monoid presentation of length 11

Properties

Finite commutative monoid with 18 elements
Isomorphic to commutative presentation: 〈a | 18a=6a〉
Grothendieck group is ℤ₁₂

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹⁸ ⇒ b⁶
a ⇒ b⁶

Idempotents

2 elements

1
b¹²

Cayley table

Idempotents are shown in bold.

	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
1	1	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
b	b	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶
b²	b²	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷
b³	b³	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸
b⁴	b⁴	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹
b⁵	b⁵	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰
b⁶	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹
b⁷	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²
b⁸	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³
b⁹	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴
b¹⁰	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵
b¹¹	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶
b¹²	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷
b¹³	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶
b¹⁴	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷
b¹⁵	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸
b¹⁶	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹
b¹⁷	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰	b¹¹	b¹²	b¹³	b¹⁴	b¹⁵	b¹⁶	b¹⁷	b⁶	b⁷	b⁸	b⁹	b¹⁰

Right Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 104 total

Length:	Presentation:	Description:	Related:
8	〈a, b \| aa=1, abbba=b〉	Finite non-commutative monoid with 18 elements	14 isomorphic
9	〈a, b \| aa=1, abbbbb=b〉	Finite non-commutative monoid with 18 elements	22 isomorphic, 9 anti-isomorphic
10	〈a, b \| aaab=1, bbbbbb=1〉	Isomorphic to ℤ₁₈	33 isomorphic
10	〈a, b \| aba=a, aaab=bb〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| aba=b, baab=aa〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| aba=b, bbbb=aa〉	Finite non-commutative monoid with 18 elements
10	〈a, b \| bb=aa, aaaba=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
10	〈a, b \| aa=a, bbbbb=ab〉	Finite non-commutative monoid with 18 elements	1 isomorphic
10	〈a, b \| aa=1, abababa=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aaaa=ab, babb=b〉	Finite non-commutative monoid with 18 elements	2 isomorphic
11	〈a, b \| aab=bb, bbbba=a〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=ab, bbbb=ab〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=ab, bbbb=ba〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aab=bb, baba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=a〉	Finite commutative monoid with 18 elements
11	〈a, b \| aaa=b, bbbbbb=b〉	Finite commutative monoid with 18 elements
11	〈a, b \| aab=b, bbbba=aa〉	Finite non-commutative monoid with 18 elements
11	〈a, b \| bb=aa, ababa=aa〉	Finite non-commutative monoid with 18 elements	1 isomorphic