#22279 ⟨a, b | aaa=1, abbabb=ba⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 174 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b⁹ = b and b ⋅ 1 = b, however b⁹ ≠ 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, b⁹
1 non-trivial idempotent:
- b⁹
Order of generators:
- a: order 3
- b: index 1, period 9

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b⁹
index 1, period 3	38 elements	b³, ab³a, b³a², b⁶, ab⁶, ...
index 1, period 9	114 elements	b, ab, ba, b², ab², ...
index 1, period 19	18 elements	ab³, bab², b²ab, b³a, ab⁶a, ...

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.	b¹⁰ ⇒ b	[99]
2.	b⁹ab ⇒ ab	[104]
3.	ab⁹ ⇒ b⁹a	[106]
4.	aba ⇒ b⁴ab³	[103]
5.	ab²a ⇒ bab⁷	[96]
6.	ab⁴a ⇒ b²ab⁸	[93]
7.	ab⁵a ⇒ b⁶ab⁵	[45]
8.	a²b ⇒ b³ab⁷a	[100]
9.	ab³ab ⇒ b(b⁶a)²	[108]
10.	ab⁷ab ⇒ b⁵ab³a	[83]
11.	ab⁶ab³ ⇒ bab⁸a	[105]
12.	ab⁸ab³ ⇒ b²a²	[97]
13.	a³ ⇒ 1	[1]
14.	ab³a² ⇒ b⁵ab	[101]
15.	ab⁶a² ⇒ (b⁵ab)²	[107]
16.	ab⁷a² ⇒ b⁸ab²	[102]
17.	ab⁸a² ⇒ b⁷ab⁴	[84]

# ab:aaa=1,abbabb=ba reversed:b/a
bbbbbbbbbb=b
bbbbbbbbbab=ab
abbbbbbbbb=bbbbbbbbba
aba=bbbbabbb
abba=babbbbbbb
abbbba=bbabbbbbbbb
abbbbba=bbbbbbabbbbb
aab=bbbabbbbbbba
abbbab=bbbbbbbabbbbbba
abbbbbbbab=bbbbbabbba
abbbbbbabbb=babbbbbbbba
abbbbbbbbabbb=bbaa
aaa=1
abbbaa=bbbbbab
abbbbbbaa=bbbbbabbbbbbab
abbbbbbbaa=bbbbbbbbabb
abbbbbbbbaa=bbbbbbbabbbb

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