#1763 ⟨a, b | abaabaaab=a⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ ba²ba³b = a and a ⋅ 1 = a, however ba²ba³b ≠ 1
Enveloping group: ⟨a, b | aabab^-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 = abaab
2. d = aaa
3. e = cca
Reduction order:
- Right-to-left recursive path with deg(c) = 0; deg(e) = 1; deg(d) = deg(a) = deg(b) = 2, d < a < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	dc ⇒ ec²d	[20]
2.	de ⇒ ec²e²d	[25]
3.	c²a ⇒ e	[4]
4.	eac ⇒ a	[10]
5.	c²eae ⇒ eca	[12]
6.	c²e²ae ⇒ ea	[16]
7.	e³ae ⇒ c²d	[22]
8.	ece²aec ⇒ c⁴d	[26]
9.	ece²ae² ⇒ c⁴e²d	[27]
10.	da ⇒ ad	[6]
11.	a² ⇒ e²ae	[13]
12.	aca ⇒ eae	[11]
13.	e(ea)² ⇒ d	[15]
14.	(ae)² ⇒ e²aeca	[18]
15.	aceae ⇒ eaeca	[14]
16.	ae²ae ⇒ d	[19]
17.	ace²ae ⇒ (ea)²	[17]
18.	db ⇒ ce²aec	[33]
19.	ceab ⇒ ebac	[28]
20.	cece²aeb ⇒ ebeaec	[32]
21.	abac ⇒ ce²aeb	[23]
22.	abae ⇒ ce²aebca	[30]
23.	abeae ⇒ ce²aeba	[31]
24.	eabe²aec ⇒ cabc²d	[39]
25.	ce²aebe²aec ⇒ abc²d	[37]
26.	eabce²aec ⇒ cabe²ae	[41]
27.	ce²aebce²aec ⇒ abe²ae	[35]
28.	eabe²ae² ⇒ cabc²e²d	[40]
29.	ce²aebe²ae² ⇒ abc²e²d	[38]
30.	ece²aebce²ae² ⇒ cabed	[46]
31.	aeab ⇒ eaebac	[29]
32.	aece²aeb ⇒ eaebeaec	[34]
33.	ebe²aeb ⇒ c³	[24]
34.	abe²aeca ⇒ ce²aebce²ae²	[36]
35.	abe²aeceae ⇒ ce²aebce²ae²ca	[42]
36.	abe²aece²ae ⇒ ce²aebc(e²a)²	[43]
37.	abe²aeb ⇒ c	[21]
38.	ab(e²a)²b ⇒ ce²aebce²ae²bac	[44]
39.	eabc(e²a)²b ⇒ cabce²aecac	[48]
40.	ce²aebc(e²a)²b ⇒ abce²aecac	[47]
41.	abe²ae²ce²aeb ⇒ ce²aebce²ae²beaec	[45]
42.	eabce²ae²ce²aeb ⇒ cabce(eaec)²	[50]
43.	ce²aebce²ae²ce²aeb ⇒ abce(eaec)²	[49]

# ab:abaabaaab=a reversed:c/e/dab abaab=c,aaa=d,cca=e morph:5/0,3/0,3/0
dc=eccd
de=ecceed
cca=e
eac=a
cceae=eca
cceeae=ea
eeeae=ccd
eceeaec=ccccd
eceeaee=cccceed
da=ad
aa=eeae
aca=eae
eeaea=d
aeae=eeaeca
aceae=eaeca
aeeae=d
aceeae=eaea
db=ceeaec
ceab=ebac
ceceeaeb=ebeaec
abac=ceeaeb
abae=ceeaebca
abeae=ceeaeba
eabeeaec=cabccd
ceeaebeeaec=abccd
eabceeaec=cabeeae
ceeaebceeaec=abeeae
eabeeaee=cabcceed
ceeaebeeaee=abcceed
eceeaebceeaee=cabed
aeab=eaebac
aeceeaeb=eaebeaec
ebeeaeb=ccc
abeeaeca=ceeaebceeaee
abeeaeceae=ceeaebceeaeeca
abeeaeceeae=ceeaebceeaeea
abeeaeb=c
abeeaeeab=ceeaebceeaeebac
eabceeaeeab=cabceeaecac
ceeaebceeaeeab=abceeaecac
abeeaeeceeaeb=ceeaebceeaeebeaec
eabceeaeeceeaeb=cabceeaeceaec
ceeaebceeaeeceeaeb=abceeaeceaec