#3109 ⟨a, b, c | ab=aa, bccb=1⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 8
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ b = a² and a ⋅ a = a², however b ≠ a

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

#	Rule	Proof
1.	dc² ⇒ c²d	[7]
2.	cb ⇒ d	[3]
3.	bc² ⇒ cd	[6]
4.	dcd ⇒ c	[5]
5.	bcd ⇒ 1	[4]
6.	a²c² ⇒ acd	[9]
7.	cdb ⇒ 1	[10]
8.	dbc ⇒ 1	[13]
9.	ab ⇒ a²	[1]
10.	b²c ⇒ db	[12]
11.	a²cd ⇒ a	[8]
12.	dbd ⇒ b	[15]
13.	d²b ⇒ b²dc	[17]
14.	adb ⇒ a³c	[14]
15.	db² ⇒ b²d	[16]

# abc:ab=aa,bccb=1 reversed:c/dab cb=d morph:2/0
dcc=ccd
cb=d
bcc=cd
dcd=c
bcd=1
aacc=acd
cdb=1
dbc=1
ab=aa
bbc=db
aacd=a
dbd=b
ddb=bbdc
adb=aaac
dbb=bbd