#16549 ⟨a, b | aba=bb, bab=aaa⟩

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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 75 elements
Not cancellative, because multiplication by a⁹ is not injective:
- a⁹ ⋅ a⁸ = a⁹ and a⁹ ⋅ 1 = a⁹, however a⁸ ≠ 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.

5 element center:
- 1, a⁴, a⁸, a¹², a¹⁶
1 non-trivial idempotent:
- a¹⁶
Order of generators:
- a: index 9, period 8
- b: index 7, period 6

Histogram:

index 1, period 1	1 element	a¹⁶
index 1, period 2	1 element	a¹²
index 1, period 3	8 elements	a³ba⁷, a²ba⁸, aba⁹, ba¹⁰, a³ba⁹, ...
index 1, period 4	18 elements	ba³b, a³ba², a²ba³, ba³ba, ba³ba², ...
index 1, period 6	8 elements	a³ba³, a²ba⁴, ba⁶, a³ba⁵, a²ba⁶, ...
index 1, period 8	12 elements	a³ba⁴, a²ba⁵, aba⁶, ba⁷, a⁹, ...
index 2, period 1	1 element	a⁸
index 2, period 3	5 elements	ba²b, a³ba, a²ba², aba³, ba⁴
index 2, period 4	3 elements	a⁶, aba⁴, ba⁵
index 2, period 6	1 element	aba⁵
index 2, period 8	6 elements	a³b, a²ba, aba², ba³, a⁵, ...
index 3, period 2	1 element	a⁴
index 3, period 8	1 element	a³
index 4, period 3	3 elements	a²b, aba, ba²
index 5, period 4	3 elements	a², ab, ba
index 7, period 6	1 element	b
index 9, period 8	1 element	a

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(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹⁷ ⇒ a⁹	[14]
2.	ba¹⁴ ⇒ ba⁶	[13]
3.	a²ba¹¹ ⇒ a²ba³	[10]
4.	a³ba¹⁰ ⇒ a³ba²	[8]
5.	a⁴b ⇒ ba⁴	[5]
6.	b² ⇒ aba	[1]
7.	bab ⇒ a³	[2]
8.	ba²ba ⇒ a³b	[4]
9.	ba³ba⁸ ⇒ ba³b	[12]
10.	aba²b ⇒ ba³	[3]
11.	aba³b ⇒ ba³ba⁷	[11]

# ab:aba=bb,bab=aaa a/b
aaaaaaaaaaaaaaaaa=aaaaaaaaa
baaaaaaaaaaaaaa=baaaaaa
aabaaaaaaaaaaa=aabaaa
aaabaaaaaaaaaa=aaabaa
aaaab=baaaa
bb=aba
bab=aaa
baaba=aaab
baaabaaaaaaaa=baaab
abaab=baaa
abaaab=baaabaaaaaaa

Same cardinality

1 unique, 2 total

Σ	#	Presentation	Description	Related
11	13119	⟨a, b \| bbb=aaa, abba=b⟩	Finite non-commutative monoid with 75 elements	1 iso

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