⟨a, b | aaaa=bbb, abab=1⟩

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Monoids with two generators and two relations

#10660 ⟨a, b | aaaa=bab, bbbb=1⟩

#10683 ⟨a, b | aaab=aaa, bbbb=1⟩

Elements

14 element center:

1, a⁴, a⁸, a¹², a¹⁶, a²⁰, a²⁴, a²⁸, a³², a³⁶, a⁴⁰, a⁴⁴, a⁴⁸, a⁵²

Order of generators:

a: order 56
b: order 42

Histogram:

order 2	13 elements	ab, ba, ba³ba⁶, ba³ba²⁰, a²⁸, ...
order 3	8 elements	a³ba¹⁹, a²ba²⁰, aba²¹, ba²², a³ba³³, ...
order 4	6 elements	a¹⁴, ba³ba¹³, ba³ba²⁷, a⁴², ba³ba⁴¹, ...
order 6	8 elements	a³ba⁵, a²ba⁶, aba⁷, ba⁸, a³ba⁴⁷, ...
order 7	6 elements	a⁸, a¹⁶, a²⁴, a³², a⁴⁰, ...
order 8	12 elements	a⁷, a³ba¹², a²ba¹³, aba¹⁴, ba¹⁵, ...
order 14	78 elements	a⁴, ba³b, a³ba², a²ba³, aba⁴, ...
order 21	48 elements	a³ba, a²ba², aba³, ba⁴, a³ba³, ...
order 28	36 elements	a², a⁶, ba³ba, (ba³)², a¹⁰, ...
order 42	48 elements	b, a²b, aba, ba², a³ba⁷, ...
order 56	72 elements	a, a³, a³b, a²ba, aba², ...

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.	a⁵⁶ ⇒ 1	[17]
2.	a⁴b ⇒ ba⁴	[3]
3.	b² ⇒ aba⁵	[4]
4.	bab ⇒ a⁵⁵	[18]
5.	ba²b ⇒ a³ba⁴⁷	[13]
6.	aba³b ⇒ ba³ba⁴³	[15]

# ab:aaaa=bbb,abab=1 a/b aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa=1 aaaab=baaaa bb=abaaaaa bab=aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa baab=aaabaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa abaaab=baaabaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

Σ	#	Presentation	Description	Related
11	22271	⟨a, b \| aaa=1, ababbb=ba⟩	Finite non-commutative monoid with 336 elements

Presentation

Description

22271

⟨a, b | aaa=1, ababbb=ba⟩

Finite non-commutative monoid with 336 elements

#10664 ⟨a, b | aaaa=bbb, abab=1⟩

Quick links

Properties

Elements

Right Cayley graph

Left Cayley graph

Rewriting system

Same cardinality

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