#9737 ⟨a, b | aa=1, ababbba=b⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality
Isomorphic instances

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 56 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b⁶ = b and b ⋅ 1 = b, however b⁶ ≠ 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.

2 element center:
- 1, b⁶
1 non-trivial idempotent:
- b⁶
Order of generators:
- a: order 2
- b: index 1, period 6

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b⁶
index 1, period 2	9 elements	b³, ab²ab, ab⁶, bab⁵, b²ab⁴, ...
index 1, period 3	8 elements	b², ab²a, b⁴, (ab²)², ab⁴a, ...
index 1, period 6	18 elements	b, ab², bab, b²a, ab⁴, ...
index 1, period 9	18 elements	ab, ba, ab³, bab², b²ab, ...

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.	b⁷ ⇒ b	[9]
2.	bab⁶ ⇒ ba	[6]
3.	b⁶a ⇒ ab⁶	[12]
4.	a² ⇒ 1	[1]
5.	aba ⇒ bab³	[5]
6.	ab³a ⇒ b⁵ab	[11]
7.	ab⁵a ⇒ b³ab⁵	[14]
8.	bab²a ⇒ ab⁴ab⁵	[15]
9.	bab⁴a ⇒ ab²ab³	[7]

# ab:aa=1,ababbba=b b/a
bbbbbbb=b
babbbbbb=ba
bbbbbba=abbbbbb
aa=1
aba=babbb
abbba=bbbbbab
abbbbba=bbbabbbbb
babba=abbbbabbbbb
babbbba=abbabbb

Same cardinality

1 unique, 2 total

Σ	#	Presentation	Description	Related
11	18415	⟨a, b \| aab=a, bbbbbbb=1⟩	Finite non-commutative monoid with 56 elements	1 iso

Isomorphic instances

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

7 total

Σ	#	Presentation	Mapping
10	10007	⟨a, b \| aa=1, ababbb=ba⟩	φ(a) = a, φ(b) = b
10	10019	⟨a, b \| aa=1, abbbab=ba⟩	φ(a) = a, φ(b) = bab
10	10326	⟨a, b \| aa=1, babbb=aba⟩	φ(a) = a, φ(b) = b
11	26587	⟨a, b \| aa=1, abababba=b⟩	φ(a) = a, φ(b) = abbbbb
11	27116	⟨a, b \| aa=1, abababb=ba⟩	φ(a) = a, φ(b) = abbbbb
11	27140	⟨a, b \| aa=1, abbabab=ba⟩	φ(a) = a, φ(b) = ab
11	27728	⟨a, b \| aa=1, bababb=aba⟩	φ(a) = a, φ(b) = abbbbb

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