#2249 ⟨a, b | aababba=baa⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 10
Infinite non-cancellative non-commutative monoid
Not right cancellative, because right multiplication by a is not injective:
- a²bab² ⋅ a = ba² and ba ⋅ a = ba², however a²bab² ≠ ba
Enveloping group: ⟨a, b | aababba^-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 = aababb
2. d = bc
3. e = babb
Reduction order:
- Right-to-left recursive path with deg(a) = deg(c) = deg(d) = 0, a < c < d; deg(e) = 1; deg(b) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	a²cda ⇒ ca²	[17]
2.	cacda ⇒ da²	[18]
3.	a²cdc ⇒ cac	[19]
4.	cacdc ⇒ dac	[20]
5.	a²cd² ⇒ cad	[21]
6.	cacd² ⇒ dad	[22]
7.	a²e ⇒ c	[6]
8.	cae ⇒ d	[7]
9.	ec ⇒ cdae	[11]
10.	a²cded ⇒ d²	[31]
11.	cacded ⇒ daed	[32]
12.	ea² ⇒ cda	[13]
13.	eac ⇒ cdc	[14]
14.	edc ⇒ cda(cd)²ae	[30]
15.	ead ⇒ cd²	[16]
16.	ed² ⇒ cdacded	[33]
17.	eda² ⇒ cda(cd)²a	[23]
18.	edac ⇒ cdac(dc)²	[24]
19.	edad ⇒ cdacdcd²	[25]
20.	cd(ae)² ⇒ ed	[15]
21.	a²cde²d ⇒ ded	[40]
22.	cacde²d ⇒ dae²d	[41]
23.	eaed ⇒ cded	[29]
24.	(ed)² ⇒ cdacde²d	[35]
25.	edaed ⇒ cda(cd)²ed	[34]
26.	c(dae)²ae ⇒ e²d	[28]
27.	e³d ⇒ (cda)²(cd)²ed(ae)²	[38]
28.	eae²d ⇒ cde²d	[39]
29.	ede²d ⇒ (cdacd)²a(cd)²ed(ae)²	[43]
30.	edae²d ⇒ cda(cd)²e²d	[42]
31.	bc ⇒ d	[3]
32.	bd ⇒ dae	[9]
33.	ba² ⇒ ca	[5]
34.	bac ⇒ c²	[8]
35.	bad ⇒ cd	[10]
36.	bed ⇒ d²(ae)²	[26]
37.	baed ⇒ ced	[27]
38.	be²d ⇒ d(dae)²ae	[36]
39.	bae²d ⇒ ce²d	[37]
40.	babe ⇒ eab²	[12]
41.	bab² ⇒ e	[4]

# ab:aababba=baa reversed:acd/e/b aababb=c,bc=d,babb=e morph:6/0,2/0,4/0
aacda=caa
cacda=daa
aacdc=cac
cacdc=dac
aacdd=cad
cacdd=dad
aae=c
cae=d
ec=cdae
aacded=dd
cacded=daed
eaa=cda
eac=cdc
edc=cdacdcdae
ead=cdd
edd=cdacded
edaa=cdacdcda
edac=cdacdcdc
edad=cdacdcdd
cdaeae=ed
aacdeed=ded
cacdeed=daeed
eaed=cded
eded=cdacdeed
edaed=cdacdcded
cdaedaeae=eed
eeed=cdacdacdcdedaeae
eaeed=cdeed
edeed=cdacdcdacdacdcdedaeae
edaeed=cdacdcdeed
bc=d
bd=dae
baa=ca
bac=cc
bad=cd
bed=ddaeae
baed=ced
beed=ddaedaeae
baeed=ceed
babe=eabb
babb=e