#4987 ⟨a, b | aaaabba=baaa⟩

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:
- a⁴b² ⋅ a = ba³ and ba² ⋅ a = ba³, however a⁴b² ≠ ba²

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

#	Rule	Proof
1.	a²e ⇒ c	[13]
2.	a²cae ⇒ ea²	[22]
3.	dc ⇒ ce	[18]
4.	da² ⇒ c	[6]
5.	dac ⇒ cae	[19]
6.	dec ⇒ c²ae²	[20]
7.	dea² ⇒ c²ae	[23]
8.	deac ⇒ c²(ae)²	[21]
9.	daec ⇒ (ca)²e²	[25]
10.	daea² ⇒ (ca)²e	[24]
11.	daeac ⇒ cac(ae)²	[30]
12.	cdad ⇒ de	[9]
13.	a(ad)² ⇒ e	[4]
14.	c(ad)² ⇒ dae	[10]
15.	de² ⇒ c²aedad	[43]
16.	deae ⇒ c²ae(ad)²	[49]
17.	dae² ⇒ (ca)²edad	[41]
18.	d(ae)² ⇒ (ca)²e(ad)²	[42]
19.	d²e ⇒ cedad	[26]
20.	d²ae ⇒ ce(ad)²	[27]
21.	dade ⇒ caedad	[31]
22.	(da)²e ⇒ cae(ad)²	[32]
23.	(de)² ⇒ c²ae²dad	[33]
24.	dedae ⇒ c²ae²(ad)²	[34]
25.	deade ⇒ c²(ae)²dad	[38]
26.	deadae ⇒ c²(ae)²(ad)²	[39]
27.	daede ⇒ (ca)²e²dad	[35]
28.	(dae)² ⇒ (ca)²e²(ad)²	[40]
29.	daeade ⇒ cac(ae)²dad	[44]
30.	daeadae ⇒ cac(ae)²(ad)²	[48]
31.	bc ⇒ cad	[15]
32.	bac ⇒ a⁴de	[16]
33.	ba³ ⇒ a⁴d	[5]
34.	ba²c ⇒ a⁴dae	[17]
35.	bec ⇒ a⁴c(acae)²dad	[52]
36.	bea² ⇒ a⁴(ca)²e(ad)²	[47]
37.	beac ⇒ a⁴c(acae)²(ad)²	[53]
38.	bae ⇒ a⁴d²ad	[14]
39.	be² ⇒ a⁴(ca)²ea(dad)²	[50]
40.	beae ⇒ a⁴(ca)²e(ad)⁴	[51]
41.	bde ⇒ cad²ad	[28]
42.	bdae ⇒ daead	[29]
43.	bade ⇒ a⁴dedad	[36]
44.	badae ⇒ a⁴de(ad)²	[37]
45.	ba²de ⇒ a⁴daedad	[45]
46.	ba²dae ⇒ a⁴dae(ad)²	[46]
47.	bede ⇒ a⁴c(acae)²(dad)²	[54]
48.	bedae ⇒ a⁴c(acae)²d(ad)³	[55]
49.	beade ⇒ a⁴ca(caea)²(dad)²	[56]
50.	beadae ⇒ a⁴c(acae)²(ad)⁴	[57]
51.	b²a ⇒ d	[3]
52.	b²e ⇒ d(ad)²	[7]

# ab:aaaabba=baaa ac/ed/b bbaaa=c,bba=d,aadad=e morph:5/1,3/1,5/0
aae=c
aacae=eaa
dc=ce
daa=c
dac=cae
dec=ccaee
deaa=ccae
deac=ccaeae
daec=cacaee
daeaa=cacae
daeac=cacaeae
cdad=de
aadad=e
cadad=dae
dee=ccaedad
deae=ccaeadad
daee=cacaedad
daeae=cacaeadad
dde=cedad
ddae=ceadad
dade=caedad
dadae=caeadad
dede=ccaeedad
dedae=ccaeeadad
deade=ccaeaedad
deadae=ccaeaeadad
daede=cacaeedad
daedae=cacaeeadad
daeade=cacaeaedad
daeadae=cacaeaeadad
bc=cad
bac=aaaade
baaa=aaaad
baac=aaaadae
bec=aaaacacaeacaedad
beaa=aaaacacaeadad
beac=aaaacacaeacaeadad
bae=aaaaddad
bee=aaaacacaeadaddad
beae=aaaacacaeadadadad
bde=caddad
bdae=daead
bade=aaaadedad
badae=aaaadeadad
baade=aaaadaedad
baadae=aaaadaeadad
bede=aaaacacaeacaedaddad
bedae=aaaacacaeacaedadadad
beade=aaaacacaeacaeadaddad
beadae=aaaacacaeacaeadadadad
bba=d
bbe=dadad