#2675 ⟨a, b | abaab=aabba⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 10
Infinite non-cancellative non-commutative monoid
Element a has infinite order
Not left cancellative, because left multiplication by c is not injective:
- c ⋅ (db)² = cdc and c ⋅ dc = cdc, however (db)² ≠ dc

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 = babba
2. d = baa
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.	c(db)² ⇒ cdc	[15]
2.	d(db)² ⇒ d²c	[16]
3.	cdbdc ⇒ (cd)²b	[19]
4.	d²bdc ⇒ d²cdb	[21]
5.	a(db)² ⇒ adc	[13]
6.	adbdc ⇒ adcdb	[17]
7.	ca ⇒ babd	[6]
8.	cb²a ⇒ babc	[5]
9.	db²a ⇒ badb	[7]
10.	cdba ⇒ ba(bd)²	[11]
11.	cdb³a ⇒ babdbc	[29]
12.	d²ba ⇒ dabd	[12]
13.	d²b³a ⇒ dabc	[24]
14.	dbda ⇒ bad²	[25]
15.	cdcba ⇒ cbadc	[20]
16.	d²cba ⇒ dbadc	[22]
17.	(cd)²a ⇒ cbadbd²	[27]
18.	d²cda ⇒ dbadbd²	[28]
19.	adba ⇒ a²bd	[10]
20.	adb³a ⇒ a²bc	[9]
21.	adcba ⇒ abadc	[18]
22.	adcda ⇒ abadbd²	[26]
23.	ba² ⇒ d	[3]
24.	bab²a ⇒ c	[2]
25.	dab²a ⇒ d²b	[8]
26.	dabda ⇒ d³	[23]
27.	a²b²a ⇒ adb	[4]
28.	a²bda ⇒ ad²	[14]

# ab:abaab=aabba b/cd/a babba=c,baa=d morph:5/3,3/1
cdbdb=cdc
ddbdb=ddc
cdbdc=cdcdb
ddbdc=ddcdb
adbdb=adc
adbdc=adcdb
ca=babd
cbba=babc
dbba=badb
cdba=babdbd
cdbbba=babdbc
ddba=dabd
ddbbba=dabc
dbda=badd
cdcba=cbadc
ddcba=dbadc
cdcda=cbadbdd
ddcda=dbadbdd
adba=aabd
adbbba=aabc
adcba=abadc
adcda=abadbdd
baa=d
babba=c
dabba=ddb
dabda=ddd
aabba=adb
aabda=add