#9759 ⟨a, b | aa=1, abbbbbb=b⟩

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 22 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b¹⁰ = b and b ⋅ 1 = b, however b¹⁰ ≠ 1

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
The order of x is the least n (if it exists) such that xⁿ = 1.
The index and period of x is the least m (index) and n (period) such that x^(m+n) = x^m.

1 element center:
- 1
2 non-trivial idempotents:
- b⁵a, b¹⁰
Order of generators:
- a: order 2
- b: index 1, period 10

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	b⁵a, b¹⁰
index 1, period 2	2 elements	b⁵, b¹⁰a
index 1, period 5	8 elements	ba, b², b³a, b⁴, b⁶, ...
index 1, period 10	8 elements	b, b²a, b³, b⁴a, b⁶a, ...

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.

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
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹¹ ⇒ b	[4]
2.	ab ⇒ b⁶	[3]
3.	a² ⇒ 1	[1]

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

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

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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#9759 ⟨a, b | aa=1, abbbbbb=b⟩

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