#3761 ⟨a, b | ababaabaab=a⟩

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 d is not injective:
- d ⋅ deab = d and d ⋅ 1 = d, however deab ≠ 1

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

#	Rule	Proof
1.	dcd ⇒ d⁴ec	[29]
2.	ecd ⇒ ed³ec	[24]
3.	dce ⇒ d(de)²c	[30]
4.	ece ⇒ e(de)²c	[32]
5.	dad ⇒ e	[4]
6.	e²ad ⇒ dc	[15]
7.	d²ead ⇒ da	[14]
8.	edead ⇒ ea	[20]
9.	dae ⇒ ead	[8]
10.	eae ⇒ d²ec	[23]
11.	c²d ⇒ cd³ec	[27]
12.	c²e ⇒ c(de)²c	[28]
13.	acd ⇒ ad³ec	[31]
14.	ace ⇒ a(de)²c	[33]
15.	ca ⇒ ac	[6]
16.	cead ⇒ adec	[18]
17.	cdead ⇒ ac	[13]
18.	a² ⇒ c	[2]
19.	adead ⇒ c	[12]
20.	dcb ⇒ d⁴ea	[36]
21.	ecb ⇒ ed³ea	[37]
22.	e²ab ⇒ da	[11]
23.	d²eab ⇒ d	[10]
24.	edeab ⇒ e	[19]
25.	c²b ⇒ cd³ea	[35]
26.	acb ⇒ ad³ea	[34]
27.	ceab ⇒ adea	[17]
28.	cdeab ⇒ c	[9]
29.	eadab ⇒ d²ea	[22]
30.	adeab ⇒ a	[5]
31.	bc ⇒ da	[7]
32.	ba ⇒ d	[3]

# ab:ababaabaab=a reversed:de/ca/b aa=c,ba=d,dad=e morph:2/0,2/0,3/0
dcd=ddddec
ecd=edddec
dce=ddedec
ece=ededec
dad=e
eead=dc
ddead=da
edead=ea
dae=ead
eae=ddec
ccd=cdddec
cce=cdedec
acd=adddec
ace=adedec
ca=ac
cead=adec
cdead=ac
aa=c
adead=c
dcb=ddddea
ecb=edddea
eeab=da
ddeab=d
edeab=e
ccb=cdddea
acb=adddea
ceab=adea
cdeab=c
eadab=ddea
adeab=a
bc=da
ba=d