⟨a, b | aaaa=1, ababbbb=1⟩

Up:

Monoids with two generators and two relations

#16589 ⟨a, b | aaaa=1, aabbbbb=1⟩

#16631 ⟨a, b | aaaa=1, bbbbbbb=1⟩

Elements

20 element center:

1, b³, b⁶, b⁹, b¹², b¹⁵, b¹⁸, b²¹, b²⁴, b²⁷, b³⁰, b³³, b³⁶, b³⁹, b⁴², b⁴⁵, b⁴⁸, b⁵¹, b⁵⁴, b⁵⁷

Order of generators:

a: order 4
b: order 60

Histogram:

order 2	7 elements	a², ab²ab⁸, b³⁰, a²b³⁰, ba²b²⁹, ...
order 3	8 elements	b²⁰, a²b²⁵, ba²b²⁴, a²b³⁵, ab²ab³³, ...
order 4	32 elements	a, ab⁵, bab⁴, b²ab³, b¹⁵, ...
order 5	4 elements	b¹², b²⁴, b³⁶, b⁴⁸
order 6	8 elements	a²b⁵, ab²ab³, ba²b⁴, b¹⁰, ab²ab¹³, ...
order 8	24 elements	a²b²ab³, ab¹⁰, bab⁹, b²ab⁸, a²b²ab⁸, ...
order 10	28 elements	(ab²)², b⁶, a²b⁶, ba²b⁵, a²b¹², ...
order 12	16 elements	b⁵, a²b¹⁰, ba²b⁹, a²b²⁰, ab²ab¹⁸, ...
order 15	32 elements	a²b, ba², b⁴, b⁸, ab²ab⁷, ...
order 20	128 elements	ab², bab, b²a, b³, ab³, ...
order 30	32 elements	b², ab²ab, a²b⁷, ba²b⁶, b¹⁴, ...
order 40	96 elements	ab, ba, a²b²a, ab⁴, bab³, ...
order 60	64 elements	b, a²b², ab²a, ba²b, a²b⁴, ...

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.	b⁶⁰ ⇒ 1	[29]
2.	b³a ⇒ ab³	[9]
3.	aba ⇒ b⁵⁶	[28]
4.	bab²a ⇒ a²b⁵³	[23]
5.	b²a² ⇒ ab²ab¹⁰	[20]
6.	a³ ⇒ bab⁴	[3]
7.	ba²b²a ⇒ a²b²ab⁴¹	[25]

# ab:aaaa=1,ababbbb=1 b/a bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=1 bbba=abbb aba=bbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb babba=aabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb bbaa=abbabbbbbbbbbb aaa=babbbb baabba=aabbabbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbbb

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

Σ	#	Presentation	Mapping
11	16609	⟨a, b \| aaaa=1, abbbbab=1⟩	φ(a) = a, φ(b) = b
11	16622	⟨a, b \| aaaa=1, bababbb=1⟩	φ(a) = a, φ(b) = b
11	16628	⟨a, b \| aaaa=1, bbababb=1⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

16609

⟨a, b | aaaa=1, abbbbab=1⟩

φ(a) = a, φ(b) = b

16622

⟨a, b | aaaa=1, bababbb=1⟩

φ(a) = a, φ(b) = b

16628

⟨a, b | aaaa=1, bbababb=1⟩

φ(a) = a, φ(b) = b

#16601 ⟨a, b | aaaa=1, ababbbb=1⟩

Quick links

Properties

Elements

Right Cayley graph

Left Cayley graph

Rewriting system

Isomorphic instances

Up:	Monoids with two generators and two relations
Prev:	#16589 ⟨a, b \| aaaa=1, aabbbbb=1⟩
Next:	#16631 ⟨a, b \| aaaa=1, bbbbbbb=1⟩