#5045 ⟨a, b | aaababa=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 d is not injective:
- da²dac ⋅ d = ead and ea ⋅ d = ead, however da²dac ≠ ea

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

#	Rule	Proof
1.	da²e ⇒ ea²	[17]
2.	a³e ⇒ da²	[16]
3.	a²cd ⇒ e	[4]
4.	da²dacd ⇒ ead	[37]
5.	a³dacd ⇒ dad	[36]
6.	daca ⇒ e	[7]
7.	a²ca ⇒ d	[3]
8.	eaca ⇒ dacd	[9]
9.	dace ⇒ eacd	[10]
10.	a²ce ⇒ dacd	[8]
11.	cd² ⇒ ae²ca	[25]
12.	cad ⇒ aeca	[18]
13.	ced ⇒ ae³ca	[38]
14.	cde ⇒ ae²cd	[27]
15.	cae ⇒ aecd	[20]
16.	ce² ⇒ ae³cd	[40]
17.	cdad ⇒ aedacd	[26]
18.	ca²d ⇒ adacd	[19]
19.	cead ⇒ ae²dacd	[39]
20.	cda² ⇒ ae²	[23]
21.	ca³ ⇒ ae	[13]
22.	cea² ⇒ ae³	[29]
23.	cdae ⇒ ae²acd	[28]
24.	ca²e ⇒ aeacd	[21]
25.	ceae ⇒ ae³acd	[41]
26.	d(acd)² ⇒ eace	[35]
27.	bd² ⇒ adeca	[30]
28.	bad ⇒ adca	[12]
29.	bed ⇒ ade²ca	[42]
30.	bde ⇒ adecd	[32]
31.	bae ⇒ adcd	[14]
32.	be² ⇒ ade²cd	[44]
33.	bdad ⇒ ad²acd	[31]
34.	ba²d ⇒ ae	[24]
35.	bead ⇒ adedacd	[43]
36.	bda² ⇒ ade	[22]
37.	ba³ ⇒ ad	[5]
38.	bea² ⇒ ade²	[34]
39.	bdae ⇒ adeacd	[33]
40.	ba²e ⇒ adacd	[15]
41.	beae ⇒ ade²acd	[45]
42.	bac ⇒ cab	[6]
43.	bab ⇒ c	[2]

# ab:aaababa=baaa reversed:dae/c/b bab=c,aaca=d,aacd=e morph:3/2,4/2,4/0
daae=eaa
aaae=daa
aacd=e
daadacd=ead
aaadacd=dad
daca=e
aaca=d
eaca=dacd
dace=eacd
aace=dacd
cdd=aeeca
cad=aeca
ced=aeeeca
cde=aeecd
cae=aecd
cee=aeeecd
cdad=aedacd
caad=adacd
cead=aeedacd
cdaa=aee
caaa=ae
ceaa=aeee
cdae=aeeacd
caae=aeacd
ceae=aeeeacd
dacdacd=eace
bdd=adeca
bad=adca
bed=adeeca
bde=adecd
bae=adcd
bee=adeecd
bdad=addacd
baad=ae
bead=adedacd
bdaa=ade
baaa=ad
beaa=adee
bdae=adeacd
baae=adacd
beae=adeeacd
bac=cab
bab=c