#4485 ⟨a, b | aaabaaab=aba⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element acb has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ a²ba³b = aba and a ⋅ ba = aba, however a²ba³b ≠ ba

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

#	Rule	Proof
1.	e⁴cdc ⇒ cde	[23]
2.	decdc ⇒ cd	[12]
3.	decde ⇒ ed	[13]
4.	de³cdc ⇒ e	[22]
5.	cd² ⇒ e	[4]
6.	(cd)² ⇒ e²cdc	[14]
7.	cded ⇒ e²cde	[15]
8.	ed² ⇒ de³cde	[30]
9.	e(dc)² ⇒ dec²d	[16]
10.	edcde ⇒ deced	[17]
11.	ede²cdc ⇒ dece	[28]
12.	e²cdc²d ⇒ cde²cdc	[27]
13.	e²cdced ⇒ cde²cde	[31]
14.	e(ecdc)² ⇒ cdc²d	[25]
15.	e²cdcecde ⇒ cdced	[26]
16.	(e²cd)²c ⇒ cdecd	[32]
17.	(e²cd)²e ⇒ cde²d	[33]
18.	e²cdce³cdc ⇒ cdce	[29]
19.	ca ⇒ (dc)²	[6]
20.	ea ⇒ dcde	[9]
21.	da ⇒ ad	[7]
22.	eb ⇒ de²cdc²	[20]
23.	ecb ⇒ dec³	[19]
24.	db ⇒ ac	[8]
25.	edcb ⇒ de³c³	[21]
26.	ecdcb ⇒ cdc²	[18]
27.	e²cdc²b ⇒ c³	[24]
28.	(dc)²b ⇒ c²	[11]
29.	a² ⇒ d	[3]
30.	ab ⇒ c	[2]

# ab:aaabaaab=aba ce/d/ab ab=c,aa=d,cdd=e morph:2/1,2/0,3/0
eeeecdc=cde
decdc=cd
decde=ed
deeecdc=e
cdd=e
cdcd=eecdc
cded=eecde
edd=deeecde
edcdc=deccd
edcde=deced
edeecdc=dece
eecdccd=cdeecdc
eecdced=cdeecde
eecdcecdc=cdccd
eecdcecde=cdced
eecdeecdc=cdecd
eecdeecde=cdeed
eecdceeecdc=cdce
ca=dcdc
ea=dcde
da=ad
eb=deecdcc
ecb=deccc
db=ac
edcb=deeeccc
ecdcb=cdcc
eecdccb=ccc
dcdcb=cc
aa=d
ab=c