#1115 ⟨a, b | abaaba=bab⟩

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Properties
Rewriting system
Other submonoids of same group

Properties

Presentation has sum-of-sides 9
Infinite cancellative non-commutative monoid
Submonoid of enveloping group: ⟨a, b | aaabbb⟩

Embedding: φ(a) = ba, φ(b) = b^-1a^-1b^-1

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

Auxiliary generators:
1. c = aba
2. d = acc
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(d) = deg(a) = deg(b) = 1, d < a < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	c³d ⇒ dc³	[13]
2.	ac² ⇒ d	[3]
3.	c³a ⇒ dc	[9]
4.	bc ⇒ c²a	[6]
5.	cb ⇒ d	[5]
6.	adc ⇒ dca	[11]
7.	acd ⇒ db	[7]
8.	bd ⇒ c²ab	[10]
9.	d³ ⇒ c⁶	[15]
10.	d²a ⇒ c⁴	[12]
11.	dab ⇒ c³	[8]
12.	ad² ⇒ dcab	[14]
13.	aba ⇒ c	[2]
14.	bab ⇒ c²	[4]

# ab:abaaba=bab c/dab aba=c,acc=d morph:3/0,3/0
cccd=dccc
acc=d
ccca=dc
bc=cca
cb=d
adc=dca
acd=db
bd=ccab
ddd=cccccc
dda=cccc
dab=ccc
add=dcab
aba=c
bab=cc

Other submonoids of same group

5 unique, 14 total

Σ	#	Presentation	Description	Related
6	67	⟨a, b \| aabbba=1⟩	Infinite non-Abelian group	8 iso
6	124	⟨a, b \| bbb=aaa⟩	Infinite cancellative non-commutative monoid
7	229	⟨a, b \| ababa=bb⟩	Infinite cancellative non-commutative monoid
8	410	⟨a, b \| abbabba=b⟩	Infinite cancellative non-commutative monoid	1 iso
11	5304	⟨a, b \| abaaaba=baab⟩	Infinite cancellative non-commutative monoid

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