⟨a, b | aaaa=1, aababbb=1⟩

Up:

Monoids with two generators and two relations

#16578 ⟨a, b | aaaa=1, aabaabb=1⟩

#16589 ⟨a, b | aaaa=1, aabbbbb=1⟩

Elements

1 element center:

Order of generators:

a: order 4
b: order 16

Histogram:

order 2	17 elements	ab⁴, bab³, b²ab², b³ab, b⁴a, ...
order 4	34 elements	a, b⁴, ab⁸, bab⁷, b²ab⁶, ...
order 8	68 elements	b², ab², bab, b²a, b⁶, ...
order 16	136 elements	b, ab, ba, b³, ab³, ...
order 17	16 elements	ab¹², bab¹¹, b²ab¹⁰, b³ab⁹, b⁴ab⁸, ...

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.	b¹⁶ ⇒ 1	[27]
2.	a² ⇒ bab³	[3]
3.	aba ⇒ b⁴ab	[30]
4.	ab²a ⇒ b⁸ab¹⁴	[33]
5.	ab³a ⇒ b³ab⁴	[29]
6.	ab⁴a ⇒ b¹²	[43]
7.	ab⁵a ⇒ b¹²ab¹³	[40]
8.	ab⁶a ⇒ b²ab⁸	[38]
9.	ab⁷a ⇒ b¹⁵ab¹²	[48]
10.	ab⁸a ⇒ b¹³ab¹⁵	[45]
11.	ab⁹a ⇒ b⁷ab⁶	[50]
12.	ab¹⁰a ⇒ b⁹ab⁵	[53]
13.	ab¹¹a ⇒ b⁵ab¹⁰	[52]
14.	ab¹²a ⇒ b¹⁴ab²	[46]
15.	ab¹³a ⇒ b⁶ab¹¹	[54]
16.	ab¹⁴a ⇒ b¹¹ab⁷	[47]
17.	ab¹⁵a ⇒ b¹⁰ab⁹	[34]

# ab:aaaa=1,aababbb=1 b/a bbbbbbbbbbbbbbbb=1 aa=babbb aba=bbbbab abba=bbbbbbbbabbbbbbbbbbbbbb abbba=bbbabbbb abbbba=bbbbbbbbbbbb abbbbba=bbbbbbbbbbbbabbbbbbbbbbbbb abbbbbba=bbabbbbbbbb abbbbbbba=bbbbbbbbbbbbbbbabbbbbbbbbbbb abbbbbbbba=bbbbbbbbbbbbbabbbbbbbbbbbbbbb abbbbbbbbba=bbbbbbbabbbbbb abbbbbbbbbba=bbbbbbbbbabbbbb abbbbbbbbbbba=bbbbbabbbbbbbbbb abbbbbbbbbbbba=bbbbbbbbbbbbbbabb abbbbbbbbbbbbba=bbbbbbabbbbbbbbbbb abbbbbbbbbbbbbba=bbbbbbbbbbbabbbbbbb abbbbbbbbbbbbbbba=bbbbbbbbbbabbbbbbbbb

Isomorphic instances

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

Σ	#	Presentation	Mapping
11	16600	⟨a, b \| aaaa=1, ababbba=1⟩	φ(a) = a, φ(b) = b
11	16607	⟨a, b \| aaaa=1, abbbaab=1⟩	φ(a) = a, φ(b) = b
11	16617	⟨a, b \| aaaa=1, baababb=1⟩	φ(a) = a, φ(b) = b
11	17146	⟨a, b \| aaaa=1, baabbb=a⟩	φ(a) = a, φ(b) = abbbbbbbbbbb
11	17148	⟨a, b \| aaaa=1, bababb=a⟩	φ(a) = a, φ(b) = abbbbbbb
11	17678	⟨a, b \| aaaa=1, babbb=aa⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

16600

⟨a, b | aaaa=1, ababbba=1⟩

φ(a) = a, φ(b) = b

16607

⟨a, b | aaaa=1, abbbaab=1⟩

φ(a) = a, φ(b) = b

16617

⟨a, b | aaaa=1, baababb=1⟩

φ(a) = a, φ(b) = b

17146

⟨a, b | aaaa=1, baabbb=a⟩

φ(a) = a, φ(b) = abbbbbbbbbbb

17148

⟨a, b | aaaa=1, bababb=a⟩

φ(a) = a, φ(b) = abbbbbbb

17678

⟨a, b | aaaa=1, babbb=aa⟩

φ(a) = a, φ(b) = b

Anti-isomorphic instances

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

Σ	#	Presentation	Mapping
11	16587	⟨a, b \| aaaa=1, aabbbab=1⟩	φ(a) = a, φ(b) = b
11	16595	⟨a, b \| aaaa=1, abaabbb=1⟩	φ(a) = a, φ(b) = b
11	16620	⟨a, b \| aaaa=1, babaabb=1⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

16587

⟨a, b | aaaa=1, aabbbab=1⟩

φ(a) = a, φ(b) = b

16595

⟨a, b | aaaa=1, abaabbb=1⟩

φ(a) = a, φ(b) = b

16620

⟨a, b | aaaa=1, babaabb=1⟩

φ(a) = a, φ(b) = b

#16582 ⟨a, b | aaaa=1, aababbb=1⟩

Quick links

Properties

Elements

Right Cayley graph

Left Cayley graph

Rewriting system

Isomorphic instances

Anti-isomorphic instances

Up:	Monoids with two generators and two relations
Prev:	#16578 ⟨a, b \| aaaa=1, aabaabb=1⟩
Next:	#16589 ⟨a, b \| aaaa=1, aabbbbb=1⟩