#26619 ⟨a, b | aa=1, abbabbba=b⟩

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

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 74 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b⁸ = b and b ⋅ 1 = b, however b⁸ ≠ 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, b⁸
1 non-trivial idempotent:
- b⁸
Order of generators:
- a: order 2
- b: index 1, period 8

Histogram:

order 2	1 element	a
index 1, period 1	1 element	b⁸
index 1, period 2	9 elements	b⁴, ab⁸, bab⁷, b²ab⁶, b³ab⁵, ...
index 1, period 3	8 elements	ab⁴, bab³, b²ab², b³ab, b⁴a, ...
index 1, period 4	18 elements	b², ab², bab, b²a, b⁶, ...
index 1, period 8	36 elements	b, ab, ba, b³, ab³, ...

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.	b⁹ ⇒ b	[21]
2.	bab⁸ ⇒ ba	[17]
3.	b⁸a ⇒ ab⁸	[22]
4.	a² ⇒ 1	[1]
5.	aba ⇒ b²ab³	[5]
6.	ab²a ⇒ bab⁵	[16]
7.	ab³a ⇒ b⁶ab	[14]
8.	ab⁴a ⇒ b⁴ab⁴	[8]
9.	ab⁵a ⇒ b⁷ab²	[23]
10.	ab⁶a ⇒ b³ab⁷	[19]
11.	ab⁷a ⇒ b⁵ab⁶	[20]

# ab:aa=1,abbabbba=b b/a
bbbbbbbbb=b
babbbbbbbb=ba
bbbbbbbba=abbbbbbbb
aa=1
aba=bbabbb
abba=babbbbb
abbba=bbbbbbab
abbbba=bbbbabbbb
abbbbba=bbbbbbbabb
abbbbbba=bbbabbbbbbb
abbbbbbba=bbbbbabbbbbb

Isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
11	27147	⟨a, b \| aa=1, abbabbb=ba⟩	φ(a) = a, φ(b) = b
11	27155	⟨a, b \| aa=1, abbbabb=ba⟩	φ(a) = a, φ(b) = ab
11	27756	⟨a, b \| aa=1, bbabbb=aba⟩	φ(a) = a, φ(b) = b

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