#2255 ⟨a, b | aabbaab=aba⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by f is not injective:
- f ⋅ ge = f² and f ⋅ f = f², however ge ≠ f

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 = ba
2. d = aabc
3. e = bd
4. f = cc
5. g = ab
Reduction order:
- Left-to-right recursive path with deg(e) = deg(f) = 0, e < f; deg(d) = 1; deg(g) = deg(c) = 2, g < c; deg(b) = 3; deg(a) = 4
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	fe ⇒ ef	[39]
2.	f²d ⇒ e²df	[56]
3.	eg ⇒ f	[16]
4.	fge ⇒ f²	[35]
5.	fgd ⇒ edf	[48]
6.	fgfd ⇒ efdf	[50]
7.	efg ⇒ f²	[41]
8.	f²g ⇒ fgf	[36]
9.	dge ⇒ df	[37]
10.	dgfd ⇒ dedf	[58]
11.	dfg ⇒ dgf	[43]
12.	ece ⇒ cf²	[42]
13.	ecd ⇒ cedf	[55]
14.	ecfd ⇒ cefdf	[68]
15.	fc ⇒ cf	[10]
16.	dce² ⇒ gdf²	[72]
17.	dcd ⇒ gd²f	[65]
18.	dced ⇒ gdedf	[66]
19.	dcfd ⇒ g(df)²	[67]
20.	dcefd ⇒ gdefdf	[73]
21.	edce ⇒ fdgf	[69]
22.	fdce ⇒ edgf²	[70]
23.	g²d ⇒ df	[47]
24.	gdgf ⇒ dce	[51]
25.	dg²e ⇒ dgf	[38]
26.	dg²fd ⇒ (df)²	[57]
27.	dgfg ⇒ dg²f	[45]
28.	edg²f ⇒ fdc	[53]
29.	fdg² ⇒ edc	[64]
30.	gdc ⇒ dg²f	[44]
31.	fgc ⇒ ce	[25]
32.	dgc ⇒ gdg	[19]
33.	cge ⇒ cf	[30]
34.	cgf ⇒ ec	[28]
35.	cfg ⇒ ec	[40]
36.	ecgd ⇒ cfdf	[49]
37.	dcgd ⇒ (gd)²f	[54]
38.	c² ⇒ f	[5]
39.	cec ⇒ fgf	[33]
40.	ce²c ⇒ f(gf)²	[59]
41.	gdg² ⇒ dc	[63]
42.	f(gf)³ ⇒ ce³c	[75]
43.	dg(gf)² ⇒ gdec	[61]
44.	dg²c ⇒ gd	[27]
45.	cgc ⇒ e	[20]
46.	bd ⇒ e	[4]
47.	bgd ⇒ ce	[15]
48.	fb ⇒ bg²	[14]
49.	dgb ⇒ g²	[18]
50.	cb ⇒ bg	[12]
51.	ceb ⇒ fgbg	[34]
52.	(fg)²bg ⇒ ce²b	[62]
53.	dg²bg ⇒ gdb	[60]
54.	cgbg ⇒ eb	[29]
55.	dg²fgbg ⇒ gdeb	[74]
56.	ae ⇒ gd	[11]
57.	af ⇒ gdg	[24]
58.	age ⇒ gdg	[26]
59.	agf ⇒ dc	[21]
60.	ac ⇒ dg	[13]
61.	agc ⇒ d	[9]
62.	ab ⇒ g	[6]
63.	agbg ⇒ db	[22]
64.	fa ⇒ edg	[32]
65.	ga ⇒ dg	[23]
66.	eca ⇒ cfdg	[46]
67.	dca ⇒ g(dg)²	[52]
68.	dcea ⇒ gdfdg	[71]
69.	ba ⇒ c	[2]

# ab:aabbaab=aba ef/d/gc/b/a ba=c,aabc=d,bd=e,cc=f,ab=g morph:2/1,4/0,2/0,2/0,2/0
fe=ef
ffd=eedf
eg=f
fge=ff
fgd=edf
fgfd=efdf
efg=ff
ffg=fgf
dge=df
dgfd=dedf
dfg=dgf
ece=cff
ecd=cedf
ecfd=cefdf
fc=cf
dcee=gdff
dcd=gddf
dced=gdedf
dcfd=gdfdf
dcefd=gdefdf
edce=fdgf
fdce=edgff
ggd=df
gdgf=dce
dgge=dgf
dggfd=dfdf
dgfg=dggf
edggf=fdc
fdgg=edc
gdc=dggf
fgc=ce
dgc=gdg
cge=cf
cgf=ec
cfg=ec
ecgd=cfdf
dcgd=gdgdf
cc=f
cec=fgf
ceec=fgfgf
gdgg=dc
fgfgfgf=ceeec
dggfgf=gdec
dggc=gd
cgc=e
bd=e
bgd=ce
fb=bgg
dgb=gg
cb=bg
ceb=fgbg
fgfgbg=ceeb
dggbg=gdb
cgbg=eb
dggfgbg=gdeb
ae=gd
af=gdg
age=gdg
agf=dc
ac=dg
agc=d
ab=g
agbg=db
fa=edg
ga=dg
eca=cfdg
dca=gdgdg
dcea=gdfdg
ba=c