#20647 ⟨a, b | ab=aa, bbbbbb=b⟩

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Properties
Elements
Right Cayley graph
Left Cayley graph
Rewriting system
Same cardinality

Properties

Presentation has sum-of-sides 11
Finite non-commutative monoid with 42 elements
Not cancellative, because multiplication by a² is not injective:
- a² ⋅ a⁵ = a² and a² ⋅ 1 = a², however a⁵ ≠ 1

Elements

Elements in the center commute with all other elements.
An idempotent element x satisfies x² = x.
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
7 non-trivial idempotents:
- a⁵, ba⁴, b²a³, b³a², b⁵, b⁴a⁶, b⁵a⁵
Order of generators:
- a: index 2, period 5
- b: index 1, period 5
Histogram:

index 1, period 1 7 elements a⁵, ba⁴, b²a³, b³a², b⁵, ...

index 1, period 5 28 elements b, a², b², a³, ba², ...

index 2, period 1 1 element b⁴a

index 2, period 5 5 elements a, ba, b²a, b³a, b⁵a

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 shortlex with a < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ab ⇒ a²	[1]
2.	b⁶ ⇒ b	[2]
3.	a⁷ ⇒ a²	[3]

# ab:ab=aa,bbbbbb=b ab
ab=aa
bbbbbb=b
aaaaaaa=aa

Same cardinality

2 unique, 12 total

Σ	#	Presentation	Description	Related
10	6101	⟨a, b \| aab=a, bbbbbb=1⟩	Finite non-commutative monoid with 42 elements	3 iso
11	14981	⟨a, b \| aaa=bb, aabaab=1⟩	Finite non-Abelian group with 42 elements	7 iso

Up:	Monoids with two generators and two relations
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index 1, period 1	7 elements	a⁵, ba⁴, b²a³, b³a², b⁵, ...
index 1, period 5	28 elements	b, a², b², a³, ba², ...
index 2, period 1	1 element	b⁴a
index 2, period 5	5 elements	a, ba, b²a, b³a, b⁵a