#3767 ⟨a, b | abababaaab=a⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element c has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ (ba)³a²b = a and a ⋅ 1 = a, however (ba)³a²b ≠ 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 = aaa
2. d = ab
3. e = ddda
Reduction order:
- Right-to-left recursive path with deg(d) = 0; deg(e) = 1; deg(c) = deg(a) = deg(b) = 2, c < a < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	cd ⇒ e²d³c	[28]
2.	ce ⇒ (e²d³)²e³c	[29]
3.	d³a ⇒ e	[4]
4.	ead ⇒ a	[5]
5.	d³eae ⇒ ed²a	[11]
6.	d³e²ae ⇒ eda	[14]
7.	d³e³ae ⇒ ea	[22]
8.	e⁴ae ⇒ d³c	[21]
9.	ede³aed ⇒ d³ed³c	[31]
10.	ed²e³aed ⇒ d⁶c	[30]
11.	ede³ae² ⇒ d³e(d³e²)²ec	[33]
12.	ed²e³ae² ⇒ d⁶e²d³e³c	[32]
13.	eb ⇒ d⁴	[9]
14.	ca ⇒ ac	[6]
15.	a² ⇒ e³ae	[15]
16.	ada ⇒ e²ae	[12]
17.	ad²a ⇒ eae	[10]
18.	e³aea ⇒ c	[19]
19.	(ae)² ⇒ e³aed²a	[25]
20.	adeae ⇒ e²aed²a	[17]
21.	ad²eae ⇒ eaed²a	[13]
22.	ae²ae ⇒ e³aeda	[26]
23.	ade²ae ⇒ e²aeda	[18]
24.	ad²e²ae ⇒ eaeda	[16]
25.	ae³ae ⇒ c	[27]
26.	ade³ae ⇒ e(ea)²	[24]
27.	ad²e³ae ⇒ (ea)²	[23]
28.	cb ⇒ e³aed	[20]
29.	ab ⇒ d	[3]

# ab:abababaaab=a reversed:d/e/cab aaa=c,ab=d,ddda=e morph:3/0,2/1,4/0
cd=eedddc
ce=eedddeedddeeec
ddda=e
ead=a
dddeae=edda
dddeeae=eda
dddeeeae=ea
eeeeae=dddc
edeeeaed=dddedddc
eddeeeaed=ddddddc
edeeeaee=dddedddeedddeeec
eddeeeaee=ddddddeedddeeec
eb=dddd
ca=ac
aa=eeeae
ada=eeae
adda=eae
eeeaea=c
aeae=eeeaedda
adeae=eeaedda
addeae=eaedda
aeeae=eeeaeda
adeeae=eeaeda
addeeae=eaeda
aeeeae=c
adeeeae=eeaea
addeeeae=eaea
cb=eeeaed
ab=d