#846 ⟨a, b | ababaaab=a⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 9
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ baba³b = a and a ⋅ 1 = a, however baba³b ≠ 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 = aaa
2. d = ba
3. e = ad
Reduction order:
- Right-to-left recursive path with deg(d) = deg(e) = 0, d < e; deg(c) = deg(a) = 1, c < a; deg(b) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	dcd ⇒ d³edc	[32]
2.	ecd ⇒ ed²edc	[34]
3.	ce ⇒ (ed)²c	[31]
4.	ad ⇒ e	[4]
5.	deae ⇒ d³c	[28]
6.	e²ae ⇒ ed²c	[29]
7.	d³ae ⇒ da	[17]
8.	ed²ae ⇒ ea	[27]
9.	edaed ⇒ ae	[19]
10.	edae² ⇒ cd	[18]
11.	c²d ⇒ cd²edc	[33]
12.	acd ⇒ (ed)²c	[35]
13.	ca ⇒ ac	[7]
14.	cdae ⇒ aedc	[26]
15.	cd²ae ⇒ ac	[24]
16.	a² ⇒ edae	[12]
17.	edaea ⇒ c	[16]
18.	(ae)² ⇒ edc	[30]
19.	aedae ⇒ c	[20]
20.	dcb ⇒ d³(ed)²ae	[41]
21.	ecb ⇒ ed²(ed)²ae	[44]
22.	(ed)²aeb ⇒ a	[21]
23.	de²daeb ⇒ d³edae	[36]
24.	e³daeb ⇒ ed²edae	[39]
25.	d³edaeb ⇒ d	[14]
26.	ed²edaeb ⇒ e	[40]
27.	c²b ⇒ cd²(ed)²ae	[43]
28.	acb ⇒ (ed)³ae	[42]
29.	cdedaeb ⇒ a(ed)²ae	[38]
30.	cd²edaeb ⇒ c	[25]
31.	ae²daeb ⇒ (ed)²ae	[37]
32.	be ⇒ d²	[6]
33.	bc ⇒ dedae	[22]
34.	ba ⇒ d	[3]

# ab:ababaaab=a reversed:de/ca/b aaa=c,ba=d,ad=e morph:3/0,2/0,2/0
dcd=dddedc
ecd=eddedc
ce=ededc
ad=e
deae=dddc
eeae=eddc
dddae=da
eddae=ea
edaed=ae
edaee=cd
ccd=cddedc
acd=ededc
ca=ac
cdae=aedc
cddae=ac
aa=edae
edaea=c
aeae=edc
aedae=c
dcb=dddededae
ecb=eddededae
ededaeb=a
deedaeb=dddedae
eeedaeb=eddedae
dddedaeb=d
eddedaeb=e
ccb=cddededae
acb=edededae
cdedaeb=aededae
cddedaeb=c
aeedaeb=ededae
be=dd
bc=dedae
ba=d