#1777 ⟨a, b | ababaaaab=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 ⋅ baba⁴b = a and a ⋅ 1 = a, however baba⁴b ≠ 1
Enveloping group: ⟨a, b | aaabba^-1b⟩

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 = ab
2. d = caa
3. e = ada
Reduction order:
- Right-to-left recursive path with deg(b) = deg(d) = deg(c) = 0, b < d < c; deg(e) = 1; deg(a) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	d²eb ⇒ dbdcdc³de	[81]
2.	dbdeb ⇒ c²dc³de	[77]
3.	dcdeb ⇒ dbdc³de	[78]
4.	dbcdeb ⇒ c⁴de	[33]
5.	dc²deb ⇒ dbc²de	[51]
6.	dbc²deb ⇒ d	[46]
7.	ed ⇒ (cd)²e	[74]
8.	ec ⇒ c³de	[30]
9.	c³deb² ⇒ c	[71]
10.	d²c³debd ⇒ dbdcde	[80]
11.	dbdc³debd ⇒ c²de	[66]
12.	dcdc³debd ⇒ dbde	[65]
13.	c²dc³debd ⇒ dc³deb	[61]
14.	dbcdc³debd ⇒ ce	[62]
15.	dc³debc ⇒ c²deb	[67]
16.	c⁴debc ⇒ db	[38]
17.	c⁴debdb ⇒ dbc³debc	[45]
18.	c³debdc ⇒ eb	[41]
19.	c⁴debe ⇒ d²c³deb	[40]
20.	e²b ⇒ c³debc(cd)³c³de	[75]
21.	dc³d(eb)² ⇒ c(cd)³c³de	[76]
22.	dbc³d(eb)² ⇒ c⁴debd²c	[84]
23.	ebdeb ⇒ c³de(bd)²c³de	[79]
24.	ebcdeb ⇒ c³debdbc²de	[73]
25.	ebc²deb ⇒ e	[47]
26.	db(c³deb)² ⇒ c⁴debd	[43]
27.	c⁴debd²c³deb ⇒ dbc³debe	[70]
28.	ebdc³debd ⇒ c³de(bd)²e	[53]
29.	ebcdc³debd ⇒ c³debd²c³deb	[64]
30.	ebc³debc ⇒ c³debd²b	[49]
31.	ebc³debdb ⇒ c³debd²bc³debc	[82]
32.	ebc³debe ⇒ c³debd³c³deb	[59]
33.	ebc³debd²c³deb ⇒ c³debd²bc³debe	[83]
34.	a ⇒ c³deb	[37]

# ab:ababaaaab=a reversed:bdc/e/a ab=c,caa=d,ada=e morph:2/1,3/3,3/0
ddeb=dbdcdcccde
dbdeb=ccdcccde
dcdeb=dbdcccde
dbcdeb=ccccde
dccdeb=dbccde
dbccdeb=d
ed=cdcde
ec=cccde
cccdebb=c
ddcccdebd=dbdcde
dbdcccdebd=ccde
dcdcccdebd=dbde
ccdcccdebd=dcccdeb
dbcdcccdebd=ce
dcccdebc=ccdeb
ccccdebc=db
ccccdebdb=dbcccdebc
cccdebdc=eb
ccccdebe=ddcccdeb
eeb=cccdebccdcdcdcccde
dcccdebeb=ccdcdcdcccde
dbcccdebeb=ccccdebddc
ebdeb=cccdebdbdcccde
ebcdeb=cccdebdbccde
ebccdeb=e
dbcccdebcccdeb=ccccdebd
ccccdebddcccdeb=dbcccdebe
ebdcccdebd=cccdebdbde
ebcdcccdebd=cccdebddcccdeb
ebcccdebc=cccdebddb
ebcccdebdb=cccdebddbcccdebc
ebcccdebe=cccdebdddcccdeb
ebcccdebddcccdeb=cccdebddbcccdebe
a=cccdeb