#1364 ⟨a, b | aaababbbba=1⟩

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Properties
Rewriting system
Other submonoids of same group
Isomorphic instances

Properties

Presentation has sum-of-sides 10
Infinite non-Abelian group
Inverses of generators:
- a^-1 = a³d
- b^-1 = cbab³
- c^-1 = d
- d^-1 = c
Reduced group presentation: ⟨a, b | aaaababbbb⟩

Isomorphism: φ(a) = a, φ(b) = b

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 = aaaa
2. d = babbbb
Reduction order:
- Right-to-left recursive path with deg(c) = deg(d) = deg(a) = 0, c < d < a; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	dc ⇒ 1	[12]
2.	cd ⇒ 1	[8]
3.	ca ⇒ ac	[5]
4.	da ⇒ ad	[11]
5.	a⁴ ⇒ c	[2]
6.	bcba ⇒ cbab	[20]
7.	baba³ ⇒ d(bc)²	[22]
8.	bab²a³ ⇒ db(bc)²	[24]
9.	b⁴c ⇒ a³bab³	[16]
10.	bab³d ⇒ adb⁴	[13]
11.	b⁴ac ⇒ a³bab³a	[18]
12.	bab³ad ⇒ adb⁴a	[14]
13.	b⁴a²c ⇒ a³bab³a²	[27]
14.	bab³a²d ⇒ adb⁴a²	[26]
15.	b⁴a³ ⇒ a³db³cb	[30]
16.	bab³a³ ⇒ db³cbc	[28]
17.	bab⁴ ⇒ d	[3]

# ab:aaababbbba=1 reversed:cda/b aaaa=c,babbbb=d magic:0
dc=1
cd=1
ca=ac
da=ad
aaaa=c
bcba=cbab
babaaa=dbcbc
babbaaa=dbbcbc
bbbbc=aaababbb
babbbd=adbbbb
bbbbac=aaababbba
babbbad=adbbbba
bbbbaac=aaababbbaa
babbbaad=adbbbbaa
bbbbaaa=aaadbbbcb
babbbaaa=dbbbcbc
babbbb=d

Other submonoids of same group

1 unique, 1 total

Σ	#	Presentation	Description	Related
11	4650	⟨a, b \| aabababa=bbb⟩	Infinite cancellative non-commutative monoid

Isomorphic instances

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

3 total

Σ	#	Presentation	Mapping
10	1386	⟨a, b \| aaabbbbaba=1⟩	φ(a) = b, φ(b) = a
10	1434	⟨a, b \| aababbbbaa=1⟩	φ(a) = a, φ(b) = b
10	1545	⟨a, b \| abbbbaaaab=1⟩	φ(a) = a, φ(b) = b

Up:	Monoids with two generators and one relation
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