#7519 ⟨a, b | aaa=1, ababba=b⟩

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

Properties

Presentation has sum-of-sides 10
Finite non-commutative monoid with 339 elements
Not cancellative, because multiplication by b is not injective:
- b ⋅ b¹⁴ = b and b ⋅ 1 = b, however b¹⁴ ≠ 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.

3 element center:
- 1, b⁷, b¹⁴
1 non-trivial idempotent:
- b¹⁴
Order of generators:
- a: order 3
- b: index 1, period 14

Histogram:

order 3	2 elements	a, a²
index 1, period 1	1 element	b¹⁴
index 1, period 2	15 elements	a²b³ab, b⁷, a²b²ab³, a²bab⁵, abab⁴ab², ...
index 1, period 3	56 elements	a²b², (ab)², ab²a, ab⁴, bab³, ...
index 1, period 6	168 elements	ab, ba, a²b, aba, ab², ...
index 1, period 7	48 elements	b², b⁴, a(ab)², a²b²a, b⁶, ...
index 1, period 14	48 elements	b, b³, a²ba, b⁵, a²bab², ...

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.	b¹⁵ ⇒ b	[32]
2.	bab¹⁴ ⇒ ba	[35]
3.	b⁷a ⇒ ab⁷	[27]
4.	ba² ⇒ abab²	[4]
5.	(ba)² ⇒ ab⁵ab³	[20]
6.	bab²a ⇒ a²b	[3]
7.	bab³a ⇒ ab³ab¹¹	[42]
8.	bab⁵a ⇒ ab⁴ab⁶	[39]
9.	bab⁶a ⇒ ab²ab⁵	[36]
10.	b²ab⁴a ⇒ ab⁶ab⁴	[41]
11.	a³ ⇒ 1	[1]

# ab:aaa=1,ababba=b b/a
bbbbbbbbbbbbbbb=b
babbbbbbbbbbbbbb=ba
bbbbbbba=abbbbbbb
baa=ababb
baba=abbbbbabbb
babba=aab
babbba=abbbabbbbbbbbbbb
babbbbba=abbbbabbbbbb
babbbbbba=abbabbbbb
bbabbbba=abbbbbbabbbb
aaa=1

Isomorphic instances

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

6 total

Σ	#	Presentation	Mapping
10	8084	⟨a, b \| aaa=1, babb=aba⟩	φ(a) = a, φ(b) = abbab
11	21717	⟨a, b \| aaa=1, abaabba=b⟩	φ(a) = aa, φ(b) = abbab
11	22233	⟨a, b \| aaa=1, aababb=ba⟩	φ(a) = a, φ(b) = abbab
11	22798	⟨a, b \| aaa=1, ababb=baa⟩	φ(a) = a, φ(b) = b
11	22806	⟨a, b \| aaa=1, abbab=baa⟩	φ(a) = aa, φ(b) = aabbab
11	22832	⟨a, b \| aaa=1, baabb=aba⟩	φ(a) = aa, φ(b) = b

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	22240	⟨a, b \| aaa=1, aabbab=ba⟩	φ(a) = a, φ(b) = abbab

Up:	Monoids with two generators and two relations
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Next:	#7529 ⟨a, b \| aaa=1, abbbba=b⟩