#5341 ⟨a, b | ababaab=aaba⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by c is not injective:
- c ⋅ db = c² and c ⋅ c = c², however db ≠ c

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

#	Rule	Proof
1.	cdb ⇒ c²	[9]
2.	d²b ⇒ dc	[10]
3.	c²bc² ⇒ cd²	[17]
4.	dcbc² ⇒ d³	[15]
5.	cdc³ ⇒ c²bcd²	[18]
6.	d²c³ ⇒ dcbcd²	[21]
7.	adb ⇒ ac	[8]
8.	acbc² ⇒ ad²	[20]
9.	adc³ ⇒ acbcd²	[23]
10.	cba ⇒ abc	[6]
11.	bca ⇒ d	[3]
12.	dba ⇒ bc²	[7]
13.	c²a ⇒ cbc²	[14]
14.	dca ⇒ dbc²	[16]
15.	cd²a ⇒ c(cb)²c²	[19]
16.	d³a ⇒ d(cb)²c²	[22]
17.	aba ⇒ c	[2]
18.	aca ⇒ abc²	[12]
19.	ad²a ⇒ a(cb)²c²	[24]

# ab:ababaab=aaba b/cd/a aba=c,bca=d morph:3/1,3/3
cdb=cc
ddb=dc
ccbcc=cdd
dcbcc=ddd
cdccc=ccbcdd
ddccc=dcbcdd
adb=ac
acbcc=add
adccc=acbcdd
cba=abc
bca=d
dba=bcc
cca=cbcc
dca=dbcc
cdda=ccbcbcc
ddda=dcbcbcc
aba=c
aca=abcc
adda=acbcbcc