#24722 ⟨a, b | aa=a, bbbbbb=ab⟩

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 22 elements

Element profile

1 element center:
- 1
4 idempotent elements:
- 1, a, b¹⁰, b¹⁰a

Complete rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

Reduction order:
- Left-to-right recursive path with deg(b) = 0; deg(a) = 1

b¹¹ ⇒ b⁶
ab ⇒ b⁶
a² ⇒ a

# ab:aa=a,bbbbbb=ab b/a
bbbbbbbbbbb=bbbbbb
ab=bbbbbb
aa=a

Cayley table

Idempotents are shown in bold.

	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
1	1	a	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
a	a	a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b	b	ba	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
ba	ba	ba	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b²	b²	b²a	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b²a	b²a	b²a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b³	b³	b³a	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b³a	b³a	b³a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁴	b⁴	b⁴a	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁴a	b⁴a	b⁴a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁵	b⁵	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁵a	b⁵a	b⁵a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b⁶	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁶a	b⁶a	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a
b⁷	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁷a	b⁷a	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a
b⁸	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁸a	b⁸a	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a
b⁹	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b⁹a	b⁹a	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a
b¹⁰	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a
b¹⁰a	b¹⁰a	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a	b⁶	b⁶a	b⁷	b⁷a	b⁸	b⁸a	b⁹	b⁹a	b¹⁰	b¹⁰a

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

3 unique, 35 total

Σ	#	Presentation	Description	Related
9	3375	⟨a, b \| aa=1, ababba=b⟩	Finite non-commutative monoid with 22 elements	15 iso, 12 anti-iso
10	9759	⟨a, b \| aa=1, abbbbbb=b⟩	Finite non-commutative monoid with 22 elements	5 iso
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Finite non-commutative monoid with 22 elements

Up:	Monoids with two generators and two relations
Prev:	#24720 ⟨a, b \| aa=a, bbabbb=bb⟩
Next:	#24723 ⟨a, b \| aa=a, bbbbbb=bb⟩