#26599 ⟨a, b | aa=1, ababbbba=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 138 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	17 elements	b⁴, (bab)², bab³a, (b²a)², bab⁷, ...
index 1, period 4	34 elements	b², ab², bab, b²a, ab²a, ...
index 1, period 8	68 elements	b, ab, ba, b³, ab³, ...
index 1, period 17	16 elements	ab⁴, bab³, b²ab², b³ab, b⁴a, ...

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:
- Right-to-left recursive path with deg(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b⁹ ⇒ b	[7]
2.	b⁸ab ⇒ ab	[8]
3.	ab⁸ ⇒ b⁸a	[10]
4.	a² ⇒ 1	[1]
5.	aba ⇒ bab⁴	[4]
6.	ab⁴a ⇒ b⁷ab	[9]
7.	ab⁷a ⇒ b⁴ab⁷	[16]
8.	ab³ab ⇒ b⁷ab⁵a	[12]
9.	ab⁶ab ⇒ b(b³a)²	[15]
10.	(ab²)² ⇒ (b⁶a)²	[18]
11.	ab⁵ab⁴ ⇒ b⁷ab²a	[14]
12.	ab⁵ab²a ⇒ (b⁵a)²b	[19]
13.	ab⁵ab³a ⇒ (bab)²	[11]

# ab:aa=1,ababbbba=b reversed:b/a
bbbbbbbbb=b
bbbbbbbbab=ab
abbbbbbbb=bbbbbbbba
aa=1
aba=babbbb
abbbba=bbbbbbbab
abbbbbbba=bbbbabbbbbbb
abbbab=bbbbbbbabbbbba
abbbbbbab=bbbbabbba
abbabb=bbbbbbabbbbbba
abbbbbabbbb=bbbbbbbabba
abbbbbabba=bbbbbabbbbbab
abbbbbabbba=babbab

Isomorphic instances

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

2 total

Σ	#	Presentation	Mapping
11	27128	⟨a, b \| aa=1, ababbbb=ba⟩	φ(a) = a, φ(b) = b
11	27742	⟨a, b \| aa=1, babbbb=aba⟩	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	27159	⟨a, b \| aa=1, abbbbab=ba⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and two relations
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