〈a, b | bb=aa, ababa=aa〉

Monoid presentation of length 11

Properties

Finite non-commutative monoid with 18 elements

Completion parameters

Reduction order: left-to-right b/a

Complete rewriting system

b¹⁰ ⇒ b⁴
ab² ⇒ b⁶
b²a ⇒ b⁶
a² ⇒ b²
a(ba)² ⇒ b²

Idempotents

2 elements

1
b⁶

Cayley table

Idempotents are shown in bold.

	1	a	b	ab	ba	b²	aba	bab	b³	(ab)²	(ba)²	b⁴	b(ab)²	b⁵	b⁶	b⁷	b⁸	b⁹
1	1	a	b	ab	ba	b²	aba	bab	b³	(ab)²	(ba)²	b⁴	b(ab)²	b⁵	b⁶	b⁷	b⁸	b⁹
a	a	b²	ab	b³	aba	b⁶	b⁷	(ab)²	b⁷	b⁸	b²	b⁸	b³	b⁹	b⁴	b⁵	b⁶	b⁷
b	b	ba	b²	bab	b⁶	b³	(ba)²	b⁷	b⁴	b(ab)²	b⁵	b⁵	b⁶	b⁶	b⁷	b⁸	b⁹	b⁴
ab	ab	aba	b⁶	(ab)²	b⁴	b⁷	b²	b⁵	b⁸	b³	b⁹	b⁹	b⁴	b⁴	b⁵	b⁶	b⁷	b⁸
ba	ba	b³	bab	b⁴	(ba)²	b⁷	b⁸	b(ab)²	b⁸	b⁹	b³	b⁹	b⁴	b⁴	b⁵	b⁶	b⁷	b⁸
b²	b²	b⁶	b³	b⁷	b⁷	b⁴	b⁵	b⁸	b⁵	b⁶	b⁶	b⁶	b⁷	b⁷	b⁸	b⁹	b⁴	b⁵
aba	aba	b⁷	(ab)²	b⁸	b²	b⁵	b⁶	b³	b⁶	b⁷	b⁷	b⁷	b⁸	b⁸	b⁹	b⁴	b⁵	b⁶
bab	bab	(ba)²	b⁷	b(ab)²	b⁵	b⁸	b³	b⁶	b⁹	b⁴	b⁴	b⁴	b⁵	b⁵	b⁶	b⁷	b⁸	b⁹
b³	b³	b⁷	b⁴	b⁸	b⁸	b⁵	b⁶	b⁹	b⁶	b⁷	b⁷	b⁷	b⁸	b⁸	b⁹	b⁴	b⁵	b⁶
(ab)²	(ab)²	b²	b⁵	b³	b⁹	b⁶	b⁷	b⁴	b⁷	b⁸	b⁸	b⁸	b⁹	b⁹	b⁴	b⁵	b⁶	b⁷
(ba)²	(ba)²	b⁸	b(ab)²	b⁹	b³	b⁶	b⁷	b⁴	b⁷	b⁸	b⁸	b⁸	b⁹	b⁹	b⁴	b⁵	b⁶	b⁷
b⁴	b⁴	b⁸	b⁵	b⁹	b⁹	b⁶	b⁷	b⁴	b⁷	b⁸	b⁸	b⁸	b⁹	b⁹	b⁴	b⁵	b⁶	b⁷
b(ab)²	b(ab)²	b³	b⁶	b⁴	b⁴	b⁷	b⁸	b⁵	b⁸	b⁹	b⁹	b⁹	b⁴	b⁴	b⁵	b⁶	b⁷	b⁸
b⁵	b⁵	b⁹	b⁶	b⁴	b⁴	b⁷	b⁸	b⁵	b⁸	b⁹	b⁹	b⁹	b⁴	b⁴	b⁵	b⁶	b⁷	b⁸
b⁶	b⁶	b⁴	b⁷	b⁵	b⁵	b⁸	b⁹	b⁶	b⁹	b⁴	b⁴	b⁴	b⁵	b⁵	b⁶	b⁷	b⁸	b⁹
b⁷	b⁷	b⁵	b⁸	b⁶	b⁶	b⁹	b⁴	b⁷	b⁴	b⁵	b⁵	b⁵	b⁶	b⁶	b⁷	b⁸	b⁹	b⁴
b⁸	b⁸	b⁶	b⁹	b⁷	b⁷	b⁴	b⁵	b⁸	b⁵	b⁶	b⁶	b⁶	b⁷	b⁷	b⁸	b⁹	b⁴	b⁵
b⁹	b⁹	b⁷	b⁴	b⁸	b⁸	b⁵	b⁶	b⁹	b⁶	b⁷	b⁷	b⁷	b⁸	b⁸	b⁹	b⁴	b⁵	b⁶

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

18 unique, 103 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=a, bbbbbb=a〉	Finite 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

Other isomorphic instances

1 total

Length:	Presentation:
11	〈a, b \| bb=aa, ababa=bb〉