#12633 ⟨a, b | babb=aa, bbbb=b⟩

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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 79 elements
Not cancellative, because multiplication by a⁴ is not injective:
- a⁴ ⋅ a⁷ = a⁴ and a⁴ ⋅ 1 = a⁴, however a⁷ ≠ 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.

4 element center:
- 1, a⁷, a³ba²b², (aba)³
2 non-trivial idempotents:
- b³, a⁷
Order of generators:
- a: index 4, period 7
- b: index 1, period 3

Histogram:

index 1, period 1	2 elements	b³, a⁷
index 1, period 3	16 elements	b, b², a⁴b, a⁵b, a⁴b², ...
index 1, period 7	6 elements	a⁴, a⁵, a⁶, a⁸, a⁹, ...
index 2, period 1	2 elements	a³ba²b², (aba)³
index 2, period 3	22 elements	a³b, a²ba, aba², a³ba, a³b², ...
index 2, period 7	15 elements	a², a³, ab³a, ba²b², b²a²b, ...
index 3, period 7	2 elements	ab³, b³a
index 4, period 3	12 elements	ab, ba, a²b, aba, ab², ...
index 4, period 7	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:
- Right-to-left recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹¹ ⇒ a⁴	[19]
2.	a⁴ba ⇒ a⁶b	[15]
3.	ba³ ⇒ a²ba²	[11]
4.	bab ⇒ a²b²	[4]
5.	a³b²a ⇒ a(a²b)²	[9]
6.	a(a²b)²a ⇒ a⁴b²	[17]
7.	a²b²a² ⇒ a³b²	[7]
8.	a²(ba²)² ⇒ a(a²b)²	[12]
9.	a²b³ ⇒ a²	[6]
10.	(a²b)²ba ⇒ a³ba²b²	[18]
11.	b³a² ⇒ a²	[10]
12.	b(ba²)² ⇒ a³	[13]
13.	b⁴ ⇒ b	[2]

# ab:babb=aa,bbbb=b reversed:a/b
aaaaaaaaaaa=aaaa
aaaaba=aaaaaab
baaa=aabaa
bab=aabb
aaabba=aaabaab
aaabaaba=aaaabb
aabbaa=aaabb
aabaabaa=aaabaab
aabbb=aa
aabaabba=aaabaabb
bbbaa=aa
bbaabaa=aaa
bbbb=b

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