#4527 ⟨a, b | aaababba=baa⟩

Contents

Properties
Rewriting system

Properties

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

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 = bb
2. d = aabac
Reduction order:
- Right-to-left 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.	a³dad²a ⇒ da²	[13]
2.	a³dad³ ⇒ dad	[11]
3.	a³dcd ⇒ d²	[27]
4.	a³dad²cd ⇒ dacd	[30]
5.	ca² ⇒ ad²a	[12]
6.	cad ⇒ ad³	[10]
7.	cd² ⇒ ad²a²dcd	[29]
8.	cda² ⇒ ad²a²dad²a	[25]
9.	cdad ⇒ ad²a²dad³	[26]
10.	a³dad²bd ⇒ dabd	[33]
11.	ba² ⇒ ada	[4]
12.	bad ⇒ ad²	[9]
13.	bd² ⇒ ada²dcd	[28]
14.	bda² ⇒ (ada)²d²a	[18]
15.	bdad ⇒ (ada)²d³	[19]
16.	a³dc²d ⇒ dcd	[43]
17.	a³dad²c²d ⇒ dac²d	[44]
18.	cacd ⇒ ad²cd	[23]
19.	(cd)² ⇒ ad²a²dc²d	[46]
20.	cdacd ⇒ ad²a²dad²cd	[32]
21.	a²bac ⇒ d	[3]
22.	adabac ⇒ bd	[8]
23.	ad²abac ⇒ cd	[14]
24.	ad⁴abac ⇒ c²d	[24]
25.	bcd ⇒ ad³abac	[20]
26.	bacd ⇒ adcd	[22]
27.	bdcd ⇒ ada²dc²d	[45]
28.	bdacd ⇒ (ada)²d²cd	[31]
29.	cb ⇒ bc	[5]
30.	cabd ⇒ ad²bd	[16]
31.	cdbd ⇒ ad²a²dad³abac	[39]
32.	cdabd ⇒ ad²a²dad²bd	[35]
33.	b² ⇒ c	[2]
34.	babd ⇒ adbd	[15]
35.	(bd)² ⇒ (ada)²d³abac	[37]
36.	bdabd ⇒ (ada)²d²bd	[34]
37.	c³d ⇒ ad⁶abac	[42]
38.	cac²d ⇒ ad²c²d	[41]
39.	cdc²d ⇒ ad²a²dad⁶abac	[50]
40.	cdac²d ⇒ ad²a²dad²c²d	[48]
41.	bc²d ⇒ ad⁵abac	[36]
42.	bac²d ⇒ adc²d	[40]
43.	bdc²d ⇒ (ada)²d⁶abac	[49]
44.	bdac²d ⇒ (ada)²d²c²d	[47]
45.	a(ab)²c ⇒ db	[7]
46.	ad(ab)²c ⇒ bdb	[17]
47.	ad²(ab)²c ⇒ cdb	[21]
48.	c²db ⇒ ad⁴(ab)²c	[38]

# ab:aaababba=baa reversed:ad/cb bb=c,aabac=d morph:2/1,5/1
aaadadda=daa
aaadaddd=dad
aaadcd=dd
aaadaddcd=dacd
caa=adda
cad=addd
cdd=addaadcd
cdaa=addaadadda
cdad=addaadaddd
aaadaddbd=dabd
baa=ada
bad=add
bdd=adaadcd
bdaa=adaadadda
bdad=adaadaddd
aaadccd=dcd
aaadaddccd=daccd
cacd=addcd
cdcd=addaadccd
cdacd=addaadaddcd
aabac=d
adabac=bd
addabac=cd
addddabac=ccd
bcd=adddabac
bacd=adcd
bdcd=adaadccd
bdacd=adaadaddcd
cb=bc
cabd=addbd
cdbd=addaadadddabac
cdabd=addaadaddbd
bb=c
babd=adbd
bdbd=adaadadddabac
bdabd=adaadaddbd
cccd=addddddabac
caccd=addccd
cdccd=addaadaddddddabac
cdaccd=addaadaddccd
bccd=adddddabac
baccd=adccd
bdccd=adaadaddddddabac
bdaccd=adaadaddccd
aababc=db
adababc=bdb
addababc=cdb
ccdb=addddababc