#5246 ⟨a, b | aabbbaa=baab⟩

Contents

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite cancellative non-commutative monoid
Element ba has infinite order
Submonoid of enveloping group: ⟨a, b | aaabba^-1b^-1b^-1a^-1bb⟩
- Embedding: φ(a) = b, φ(b) = a

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

#	Rule	Proof
1.	bf ⇒ fe	[16]
2.	bd ⇒ fc	[7]
3.	be ⇒ fd	[12]
4.	db ⇒ e	[4]
5.	cb ⇒ d	[3]
6.	eb ⇒ f	[5]
7.	dfd ⇒ e²	[14]
8.	dfc ⇒ ed	[10]
9.	dfe ⇒ ef	[18]
10.	cf² ⇒ dfb	[23]
11.	cfd ⇒ de	[13]
12.	cfc ⇒ d²	[9]
13.	cfe ⇒ df	[17]
14.	ef² ⇒ f²b	[25]
15.	efb ⇒ df²	[21]
16.	efd ⇒ fe	[15]
17.	efc ⇒ fd	[11]
18.	efe ⇒ f²	[19]
19.	df³ ⇒ f³d	[28]
20.	df²d ⇒ f²bc	[26]
21.	df²e ⇒ f³c	[27]
22.	ca ⇒ ac	[8]
23.	cfaf ⇒ d²ab³	[33]
24.	cfad ⇒ d²ab	[29]
25.	cfac ⇒ d²a	[22]
26.	cfae ⇒ d²ab²	[31]
27.	efaf ⇒ fdab³	[34]
28.	efad ⇒ fdab	[30]
29.	efac ⇒ fda	[24]
30.	efae ⇒ fdab²	[32]
31.	eda ⇒ dfac	[20]
32.	a² ⇒ c	[2]

# ab:aabbbaa=baab fbdce/a aa=c,cb=d,db=e,eb=f morph:2/1,2/1,2/1,2/0
bf=fe
bd=fc
be=fd
db=e
cb=d
eb=f
dfd=ee
dfc=ed
dfe=ef
cff=dfb
cfd=de
cfc=dd
cfe=df
eff=ffb
efb=dff
efd=fe
efc=fd
efe=ff
dfff=fffd
dffd=ffbc
dffe=fffc
ca=ac
cfaf=ddabbb
cfad=ddab
cfac=dda
cfae=ddabb
efaf=fdabbb
efad=fdab
efac=fda
efae=fdabb
eda=dfac
aa=c