#5298 ⟨a, b | abaaaba=aaab⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by c is not injective:
- c ⋅ cb = ce and c ⋅ e = ce, however cb ≠ e

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

#	Rule	Proof
1.	cfce ⇒ c²f	[41]
2.	cfec ⇒ c²e	[40]
3.	efce ⇒ ecf	[33]
4.	efec ⇒ ece	[38]
5.	f²ce ⇒ fcf	[34]
6.	f²ec ⇒ fce	[39]
7.	dfce ⇒ dcf	[32]
8.	dfec ⇒ dce	[35]
9.	ed ⇒ df	[29]
10.	fd ⇒ def	[30]
11.	c²d ⇒ cdec	[23]
12.	ecd ⇒ dce	[10]
13.	fcd ⇒ dece	[44]
14.	d²ce ⇒ f	[5]
15.	d²ec ⇒ e	[20]
16.	dcd ⇒ e	[4]
17.	afce ⇒ acf	[43]
18.	afec ⇒ ace	[42]
19.	acd ⇒ adec	[22]
20.	ca ⇒ ac	[8]
21.	ea ⇒ dec	[19]
22.	fa ⇒ de²c	[24]
23.	a² ⇒ c	[2]
24.	bc ⇒ da	[9]
25.	ba ⇒ d	[3]
26.	eb ⇒ d²e²	[26]
27.	fb ⇒ d²fe²	[31]
28.	c²b ⇒ ce	[13]
29.	ecb ⇒ e²	[17]
30.	fcb ⇒ fe	[25]
31.	dcb ⇒ de	[14]
32.	acb ⇒ ae	[12]

# ab:abaaaba=aaab cef/d/a/b aa=c,ba=d,dcd=e,ddce=f morph:2/0,2/1,3/0,4/0
cfce=ccf
cfec=cce
efce=ecf
efec=ece
ffce=fcf
ffec=fce
dfce=dcf
dfec=dce
ed=df
fd=def
ccd=cdec
ecd=dce
fcd=dece
ddce=f
ddec=e
dcd=e
afce=acf
afec=ace
acd=adec
ca=ac
ea=dec
fa=deec
aa=c
bc=da
ba=d
eb=ddee
fb=ddfee
ccb=ce
ecb=ee
fcb=fe
dcb=de
acb=ae