#21731 ⟨a, b | aaa=1, ababbbb=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 111 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b¹² = b and b ⋅ 1 = b, however b¹² ≠ 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
3 non-trivial idempotents:
- bab³a, b¹², ab¹²a²
Order of generators:
- a: order 3
- b: index 1, period 12

Histogram:

order 3	2 elements	a, a²
index 1, period 1	3 elements	bab³a, b¹², ab¹²a²
index 1, period 2	3 elements	b⁶, ab⁶a², bab⁹a
index 1, period 3	24 elements	b⁴, ab⁴, bab³, b⁴a, ab⁴a, ...
index 1, period 4	18 elements	aba, ba², b³, ab³a², ab⁵, ...
index 1, period 6	24 elements	b², ab², bab, b²a, ab²a, ...
index 1, period 12	36 elements	b, ab, ba, aba², 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.	b¹³ ⇒ b	[13]
2.	b²ab ⇒ ab³	[10]
3.	a²b ⇒ bab⁴	[3]
4.	(ab)² ⇒ b¹⁰	[12]
5.	a³ ⇒ 1	[1]

# ab:aaa=1,ababbbb=b b/a
bbbbbbbbbbbbb=b
bbab=abbb
aab=babbbb
abab=bbbbbbbbbb
aaa=1

Isomorphic instances

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

1 total

Σ	#	Presentation	Mapping
11	21745	⟨a, b \| aaa=1, abbbabb=b⟩	φ(a) = a, φ(b) = b

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