#26601 ⟨a, b | aa=1, ababbbbb=b⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Isomorphic instances
Anti-isomorphic instances

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 62 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b¹⁵ = b and b ⋅ 1 = b, however b¹⁵ ≠ 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:
- b¹⁵, ab¹⁵a
Order of generators:
- a: order 2
- b: index 1, period 15

Histogram:

order 2	1 element	a
index 1, period 1	2 elements	b¹⁵, ab¹⁵a
index 1, period 2	6 elements	ab⁵, b⁵a, ab¹⁰, b¹⁰a, ab¹⁵, ...
index 1, period 3	4 elements	b⁵, ab⁵a, b¹⁰, ab¹⁰a
index 1, period 5	8 elements	b³, ab³a, b⁶, ab⁶a, b⁹, ...
index 1, period 10	24 elements	ab, ba, ab², b²a, ab³, ...
index 1, period 15	16 elements	b, b², aba, ab²a, b⁴, ...

Right Cayley graph

Left Cayley graph

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.	b¹⁶ ⇒ b	[11]
2.	bab ⇒ ab¹²	[9]
3.	a² ⇒ 1	[1]

# ab:aa=1,ababbbbb=b b/a
bbbbbbbbbbbbbbbb=b
bab=abbbbbbbbbbbb
aa=1

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	26633	⟨a, b \| aa=1, abbbbabb=b⟩	φ(a) = a, φ(b) = b
11	27217	⟨a, b \| aa=1, babbbbb=ab⟩	φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	27232	⟨a, b \| aa=1, bbabbbb=ba⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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Next:	#26619 ⟨a, b \| aa=1, abbabbba=b⟩