#22315 ⟨a, b | aaa=1, babbbb=ab⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Anti-isomorphic instances

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 255 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b²⁸ = b and b ⋅ 1 = b, however b²⁸ ≠ 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:
- b²⁸, a²b²⁸a, ab²⁸a²
Order of generators:
- a: order 3
- b: index 1, period 28

Histogram:

order 3	2 elements	a, a²
index 1, period 1	3 elements	b²⁸, a²b²⁸a, ab²⁸a²
index 1, period 2	3 elements	b¹⁴, a²b¹⁴a, ab¹⁴a²
index 1, period 3	42 elements	ab⁴, b⁴a, a²b⁴, ab⁴a, b⁴a², ...
index 1, period 4	6 elements	b⁷, a²b⁷a, ab⁷a², b²¹, a²b²¹a, ...
index 1, period 6	42 elements	ab², b²a, a²b², ab²a, b²a², ...
index 1, period 7	18 elements	b⁴, a²b⁴a, ab⁴a², b⁸, a²b⁸a, ...
index 1, period 12	84 elements	ab, ba, a²b, aba, ba², ...
index 1, period 14	18 elements	b², a²b²a, ab²a², b⁶, a²b⁶a, ...
index 1, period 28	36 elements	b, b³, a²ba, aba², b⁵, ...

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.	b²⁹ ⇒ b	[18]
2.	bab ⇒ ab²⁶	[16]
3.	ba²b ⇒ a²b¹⁰	[6]
4.	a³ ⇒ 1	[1]

# ab:aaa=1,babbbb=ab b/a
bbbbbbbbbbbbbbbbbbbbbbbbbbbbb=b
bab=abbbbbbbbbbbbbbbbbbbbbbbbbb
baab=aabbbbbbbbbb
aaa=1

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	22323	⟨a, b \| aaa=1, bbabbb=ba⟩	φ(a) = aa, φ(b) = b

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