#5196 ⟨a, b | aababba=abab⟩

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 d is not injective:
- d ⋅ bc = d²ba and d ⋅ dba = d²ba, however bc ≠ dba

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

#	Rule	Proof
1.	dc ⇒ cd	[7]
2.	c² ⇒ d	[3]
3.	ea ⇒ a³de	[22]
4.	ed ⇒ (ad)²e	[19]
5.	ec ⇒ ade	[18]
6.	ab ⇒ c	[2]
7.	adba ⇒ d	[6]
8.	ad²ba ⇒ db	[14]
9.	d³ba ⇒ ae	[13]
10.	adbd ⇒ d²ba	[9]
11.	d²bd ⇒ a²de	[20]
12.	dbc ⇒ d²ba	[10]
13.	d²bac ⇒ dbd	[12]
14.	eb ⇒ dba²e	[17]
15.	db² ⇒ a²e	[15]
16.	(db)² ⇒ e	[4]
17.	dbd²ba ⇒ ade	[21]

# ab:aababba=abab reversed:a/dc/eb ab=c,cc=d,dbdb=e morph:2/0,2/0,4/0
dc=cd
cc=d
ea=aaade
ed=adade
ec=ade
ab=c
adba=d
addba=db
dddba=ae
adbd=ddba
ddbd=aade
dbc=ddba
ddbac=dbd
eb=dbaae
dbb=aae
dbdb=e
dbddba=ade