#2595 ⟨a, b | ababab=abba⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not left cancellative, because left multiplication by a is not injective:
- a ⋅ b(ab)² = ab²a and a ⋅ b²a = ab²a, however b(ab)² ≠ b²a
Enveloping group: ⟨a, b | aaba^-1b^-1⟩

Rewriting system

Format:	Pretty Plain
Word to reduce:
	Tips: Lowercase letters stand for generators. Spaces are ignored. Numbers repeat the previous letter, e.g. `b90`.
Reduction strategy:	Leftmost Rightmost
Path to normal form:	1 1

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

#	Rule	Proof
1.	c³b ⇒ cbc	[37]
2.	cbc²b ⇒ cb²c	[32]
3.	dbc²b ⇒ db²c	[43]
4.	cdc²b ⇒ cdbc	[38]
5.	cdcb²c²b ⇒ cdcb³c	[47]
6.	cbdc²b ⇒ cbdbc	[40]
7.	c²d ⇒ cdc	[16]
8.	bcbd ⇒ db²c	[9]
9.	cbcd ⇒ cbdc	[25]
10.	c²bdcb²c²b ⇒ c²bdcb³c	[51]
11.	bc²bd ⇒ d(bc)²	[28]
12.	cb²cd ⇒ cd(bc)²	[42]
13.	d²bc ⇒ bcd	[14]
14.	d²b²c ⇒ bcdcb	[31]
15.	dbd ⇒ bc	[7]
16.	dbcd ⇒ bc²	[20]
17.	db²cd ⇒ bcdbc	[24]
18.	cd²b ⇒ cbd	[13]
19.	cd²cb ⇒ c²bd	[22]
20.	cd²c²b ⇒ cbdc	[39]
21.	(cd)² ⇒ cd²c	[18]
22.	cdcbd ⇒ c²bc	[19]
23.	cdcb²d ⇒ c²bdcb	[46]
24.	cdcb³cd ⇒ c²(bc)³	[48]
25.	bcd² ⇒ dbc²	[27]
26.	cbd² ⇒ cdbc	[23]
27.	cbdcd ⇒ cdbc²	[33]
28.	(cbd)² ⇒ c(bc)²	[41]
29.	c²bdcb²d ⇒ cdcb³c	[49]
30.	c²bdcb³cd ⇒ c²(bc)⁴	[50]
31.	d³b ⇒ bc	[10]
32.	cd³ ⇒ c³	[35]
33.	cd²cd ⇒ c⁴	[36]
34.	ab²c ⇒ cbd	[8]
35.	abc² ⇒ cd²	[26]
36.	a(bc)² ⇒ c²bd	[21]
37.	abcb²c ⇒ c²bdcb	[45]
38.	adbc ⇒ cd	[12]
39.	adb²c ⇒ cdcb	[30]
40.	abd ⇒ c	[6]
41.	abcd ⇒ c²	[15]
42.	ad²b ⇒ c	[5]
43.	ba ⇒ d	[3]
44.	ca ⇒ ad³	[11]
45.	cda ⇒ ad⁵	[17]
46.	cbda ⇒ c³	[44]
47.	cd²a ⇒ ad⁷	[34]

# ab:ababab=abba bc/d/a abba=c,ba=d morph:4/0,2/2
cccb=cbc
cbccb=cbbc
dbccb=dbbc
cdccb=cdbc
cdcbbccb=cdcbbbc
cbdccb=cbdbc
ccd=cdc
bcbd=dbbc
cbcd=cbdc
ccbdcbbccb=ccbdcbbbc
bccbd=dbcbc
cbbcd=cdbcbc
ddbc=bcd
ddbbc=bcdcb
dbd=bc
dbcd=bcc
dbbcd=bcdbc
cddb=cbd
cddcb=ccbd
cddccb=cbdc
cdcd=cddc
cdcbd=ccbc
cdcbbd=ccbdcb
cdcbbbcd=ccbcbcbc
bcdd=dbcc
cbdd=cdbc
cbdcd=cdbcc
cbdcbd=cbcbc
ccbdcbbd=cdcbbbc
ccbdcbbbcd=ccbcbcbcbc
dddb=bc
cddd=ccc
cddcd=cccc
abbc=cbd
abcc=cdd
abcbc=ccbd
abcbbc=ccbdcb
adbc=cd
adbbc=cdcb
abd=c
abcd=cc
addb=c
ba=d
ca=addd
cda=addddd
cbda=ccc
cdda=addddddd