#17655 ⟨a, b | aaaa=1, ababb=ba⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element c has infinite order
Not left cancellative, because left multiplication by e is not injective:
- e ⋅ d² = e and e ⋅ 1 = e, however d² ≠ 1
Not right cancellative, because right multiplication by e is not injective:
- d² ⋅ e = e and 1 ⋅ e = e, however d² ≠ 1

Rewriting system

Format:	Pretty Plain
Word:
	Enter a word above to compute its normal form. Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Strategy:	Leftmost Rightmost
Result:	1 1

Auxiliary generators:
1. c = abb
2. d = bbbb
3. e = daa
Reduction order:
- Left-to-right recursive path with deg(e) = deg(d) = 0, e < d; deg(c) = 1; deg(a) = deg(b) = 2, a < b
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	ede ⇒ d	[67]
2.	ed² ⇒ e	[66]
3.	d²e ⇒ e	[70]
4.	d³ ⇒ d	[69]
5.	cd² ⇒ c	[73]
6.	d²c ⇒ c	[96]
7.	dcdec ⇒ cdeced	[133]
8.	e²cdc ⇒ d(cd)²e	[140]
9.	decdc ⇒ e(cd)²e	[141]
10.	dc²ec ⇒ ec²ecd	[114]
11.	dcec² ⇒ ecec²d	[144]
12.	dcecdc ⇒ cd(ec)²de	[87]
13.	e²c²dec ⇒ dcdc²ed	[154]
14.	dec²ec ⇒ c²ecd	[113]
15.	dec²dec ⇒ ecdc²ed	[155]
16.	decec² ⇒ cec²d	[146]
17.	ec²ecdc ⇒ cec²dece	[136]
18.	e(cec)² ⇒ c²ec²ded	[132]
19.	dc³ec ⇒ cdecedcecd	[161]
20.	e(ec)³c ⇒ d(cd)²e²c²d	[147]
21.	d(ec)³c ⇒ e(cd)²e²c²d	[148]
22.	c²ec²dec ⇒ d	[54]
23.	d(ce)³c² ⇒ cd(ec)²de²c²d	[145]
24.	e²c⁴ec ⇒ dcdc²edcecd	[166]
25.	e²c³ec² ⇒ dcd(c²d)²	[163]
26.	dec⁴ec ⇒ ecdc²edcecd	[168]
27.	dec³ec² ⇒ ecd(c²d)²	[165]
28.	ec⁴ec² ⇒ cec²d(ec)³de	[170]
29.	ec(ce)³c² ⇒ cec²dece²c²d	[173]
30.	dc⁴ec² ⇒ e(c²ec)²de	[171]
31.	c²ec⁴ec ⇒ dcecd	[149]
32.	c(cec²)² ⇒ dec²d	[116]
33.	dc⁵ec² ⇒ cd(ece)²c²dcecde	[174]
34.	e²c⁶ec² ⇒ dcdc²e²cec²dcecde	[176]
35.	dec⁶ec² ⇒ ecdc²e²cec²dcecde	[177]
36.	c²ec⁶ec² ⇒ ecec²dcecde	[175]
37.	ea ⇒ aec(dc²)²e	[164]
38.	da ⇒ aecd(c²d)²	[172]
39.	ca ⇒ ac²ec²	[92]
40.	bd² ⇒ b	[71]
41.	eb ⇒ bc(d(ec)²)²	[153]
42.	db ⇒ bd	[15]
43.	cb ⇒ bcec	[49]
44.	a²e ⇒ de²	[108]
45.	a²d ⇒ ded	[107]
46.	a²c ⇒ dec	[109]
47.	ba ⇒ abc	[6]
48.	b² ⇒ adec	[99]
49.	a²b ⇒ bc³ec²de	[126]
50.	a⁴ ⇒ 1	[1]

# ab:aaaa=1,ababb=ba ed/c/ab abb=c,bbbb=d,daa=e custom:0
ede=d
edd=e
dde=e
ddd=d
cdd=c
ddc=c
dcdec=cdeced
eecdc=dcdcde
decdc=ecdcde
dccec=eccecd
dcecc=ececcd
dcecdc=cdececde
eeccdec=dcdcced
deccec=ccecd
deccdec=ecdcced
dececc=ceccd
eccecdc=ceccdece
ececcec=cceccded
dcccec=cdecedcecd
eecececc=dcdcdeeccd
decececc=ecdcdeeccd
cceccdec=d
dcecececc=cdececdeeccd
eeccccec=dcdccedcecd
eecccecc=dcdccdccd
deccccec=ecdccedcecd
decccecc=ecdccdccd
eccccecc=ceccdecececde
eccecececc=ceccdeceeccd
dccccecc=eccecccecde
cceccccec=dcecd
ccecccecc=deccd
dcccccecc=cdeceececcdcecde
eeccccccecc=dcdcceececcdcecde
deccccccecc=ecdcceececcdcecde
cceccccccecc=ececcdcecde
ea=aecdccdcce
da=aecdccdccd
ca=accecc
bdd=b
eb=bcdececdecec
db=bd
cb=bcec
aae=dee
aad=ded
aac=dec
ba=abc
bb=adec
aab=bccceccde
aaaa=1