#5166 ⟨a, b | aababaa=aaab⟩

Contents

Properties
Rewriting system

Properties

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

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

#	Rule	Proof
1.	d³a² ⇒ dad	[21]
2.	dad²a² ⇒ da²d	[31]
3.	a³d²a² ⇒ a⁴d	[40]
4.	da²d²a² ⇒ da³d	[45]
5.	ca² ⇒ d	[3]
6.	cad²a² ⇒ d²	[25]
7.	dc ⇒ cad²	[24]
8.	dac ⇒ d³	[20]
9.	da²c ⇒ dad²	[44]
10.	a⁴c ⇒ a³d²	[34]
11.	da³c ⇒ da²d²	[37]
12.	db ⇒ cad	[14]
13.	dab ⇒ d²	[19]
14.	a³bad²a² ⇒ a³ba²d	[47]
15.	a³ba²d²a² ⇒ a³ba³d	[48]
16.	da²b ⇒ dad	[17]
17.	a⁴b ⇒ a³d	[28]
18.	da³b ⇒ da²d	[43]
19.	cabad²a² ⇒ caba²d	[46]
20.	caba²d²a² ⇒ caba³d	[49]
21.	a³caba² ⇒ a³bad	[29]
22.	a³bac ⇒ a³cab	[27]
23.	a³ba²c ⇒ a³bad²	[35]
24.	a³ba³c ⇒ a³ba²d²	[36]
25.	a(ab)²a² ⇒ a³b	[1]
26.	a³bab ⇒ a³c	[4]
27.	a(a²b)² ⇒ a³bad	[41]
28.	(a³b)² ⇒ a³ba²d	[30]
29.	(ca)²ba² ⇒ cabad	[32]
30.	cabac ⇒ (ca)²b	[13]
31.	caba²c ⇒ cabad²	[38]
32.	caba³c ⇒ caba²d²	[39]
33.	cbaba² ⇒ cab	[6]
34.	c(ab)² ⇒ cac	[11]
35.	caba²b ⇒ cabad	[42]
36.	caba³b ⇒ caba²d	[33]
37.	b²aba² ⇒ c	[2]

# ab:aababaa=aaab ad/cb bbabaa=c,caa=d morph:6/0,3/0
dddaa=dad
daddaa=daad
aaaddaa=aaaad
daaddaa=daaad
caa=d
caddaa=dd
dc=cadd
dac=ddd
daac=dadd
aaaac=aaadd
daaac=daadd
db=cad
dab=dd
aaabaddaa=aaabaad
aaabaaddaa=aaabaaad
daab=dad
aaaab=aaad
daaab=daad
cabaddaa=cabaad
cabaaddaa=cabaaad
aaacabaa=aaabad
aaabac=aaacab
aaabaac=aaabadd
aaabaaac=aaabaadd
aababaa=aaab
aaabab=aaac
aaabaab=aaabad
aaabaaab=aaabaad
cacabaa=cabad
cabac=cacab
cabaac=cabadd
cabaaac=cabaadd
cbabaa=cab
cabab=cac
cabaab=cabad
cabaaab=cabaad
bbabaa=c