#9759 ⟨a, b | aa=1, abbbbbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 22 elements

Element profile

1 element center:
- 1
3 idempotent elements:
- 1, b⁵a, b¹⁰

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 ⇒ b
ab ⇒ b⁶
a² ⇒ 1

# ab:aa=1,abbbbbb=b b/a
bbbbbbbbbbb=b
ab=bbbbbb
aa=1

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

Right Cayley graph

Idempotents are shown in bold.

Left Cayley graph

Idempotents are shown in bold.

Others with same cardinality

3 unique, 30 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
11	19756	⟨a, b \| aba=b, baaab=aa⟩	Finite non-commutative monoid with 22 elements
11	24722	⟨a, b \| aa=a, bbbbbb=ab⟩	Finite non-commutative monoid with 22 elements

Other isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

5 total

Σ	#	Presentation	Mapping
10	10062	⟨a, b \| aa=1, bbbbbb=ab⟩	φ(a) = a, φ(b) = b
11	26621	⟨a, b \| aa=1, abbabbbb=b⟩	φ(a) = a, φ(b) = ba
11	26629	⟨a, b \| aa=1, abbbabbb=b⟩	φ(a) = a, φ(b) = ba
11	27231	⟨a, b \| aa=1, bbabbbb=ab⟩	φ(a) = a, φ(b) = ba
11	27235	⟨a, b \| aa=1, bbbabbb=ab⟩	φ(a) = a, φ(b) = ba

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