#5740 ⟨a, b | aabbaa=baabb⟩

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Properties
Rewriting system

Properties

Presentation has sum-of-sides 11
Infinite cancellative non-commutative monoid
Submonoid of enveloping group: ⟨a, b | aabbaab^-1a^-1a^-1b^-1b^-1⟩

Anti-isomorphic embedding: φ(a) = a, φ(b) = b

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 = baa
Reduction order:
- Right-to-left shortlex with a < c < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	ba² ⇒ c	[2]
2.	a²bc ⇒ cb²	[4]
3.	bcb² ⇒ cbc	[5]
4.	cbca² ⇒ (bc)²	[8]
5.	bacb² ⇒ cabc	[6]
6.	cb⁴ ⇒ a²cbc	[7]
7.	cabca² ⇒ bacbc	[12]
8.	cbc²b² ⇒ (bc)³	[9]
9.	a²b(bc)² ⇒ cb³ca²	[10]
10.	cbcacb² ⇒ (bc)²abc	[11]
11.	cabc²b² ⇒ bac(bc)²	[13]
12.	a²b²acbc ⇒ cb²abca²	[15]
13.	cabcacb² ⇒ bacbcabc	[14]
14.	cb³ca⁴ ⇒ a²b³cbc	[17]
15.	cb³ca²b² ⇒ a²b²c²bc	[16]
16.	cb²abca⁴ ⇒ a²b²a(bc)²	[21]
17.	cb²abca²b² ⇒ a²b²ac²bc	[20]
18.	cb³ca²cb² ⇒ a²b²(bc)³	[18]
19.	cb³ca³cb² ⇒ a²b³cbcabc	[19]
20.	cb²abca²cb² ⇒ a²b²a(bc)³	[22]
21.	(cb³c)²a² ⇒ a²b²(c²b)²c	[24]
22.	cb²abca³cb² ⇒ a²b²a(bc)²abc	[23]
23.	cb³c²b²abca² ⇒ a²b²c²bcacbc	[25]
24.	cb²abc²b³ca² ⇒ a²b²a(c²b)²c	[26]
25.	c(b³c²)²b² ⇒ a²b(bc²)²(bc)²	[28]
26.	(cb²abc)²a² ⇒ a²b²ac²bcacbc	[27]
27.	(cb³c)²acb² ⇒ a²b²(c²b)²cabc	[29]
28.	cb³c²b²abc²b² ⇒ a²b²c²bcac(bc)²	[30]
29.	cb²a(bc²b²)² ⇒ a²b²a(c²b)²cbc	[32]
30.	cb³c²b²abcacb² ⇒ a²b²c(cbca)²bc	[31]
31.	cb²abc²b³cacb² ⇒ a²b²a(c²b)²cabc	[33]
32.	cb²(abc²b²)² ⇒ a²b²ac²bcac(bc)²	[34]
33.	(cb²abc)²acb² ⇒ a²b²ac(cbca)²bc	[35]

# ab:aabbaa=baabb reversed:acb baa=c morph:3/0
baa=c
aabc=cbb
bcbb=cbc
cbcaa=bcbc
bacbb=cabc
cbbbb=aacbc
cabcaa=bacbc
cbccbb=bcbcbc
aabbcbc=cbbbcaa
cbcacbb=bcbcabc
cabccbb=bacbcbc
aabbacbc=cbbabcaa
cabcacbb=bacbcabc
cbbbcaaaa=aabbbcbc
cbbbcaabb=aabbccbc
cbbabcaaaa=aabbabcbc
cbbabcaabb=aabbaccbc
cbbbcaacbb=aabbbcbcbc
cbbbcaaacbb=aabbbcbcabc
cbbabcaacbb=aabbabcbcbc
cbbbccbbbcaa=aabbccbccbc
cbbabcaaacbb=aabbabcbcabc
cbbbccbbabcaa=aabbccbcacbc
cbbabccbbbcaa=aabbaccbccbc
cbbbccbbbccbb=aabbccbccbcbc
cbbabccbbabcaa=aabbaccbcacbc
cbbbccbbbcacbb=aabbccbccbcabc
cbbbccbbabccbb=aabbccbcacbcbc
cbbabccbbbccbb=aabbaccbccbcbc
cbbbccbbabcacbb=aabbccbcacbcabc
cbbabccbbbcacbb=aabbaccbccbcabc
cbbabccbbabccbb=aabbaccbcacbcbc
cbbabccbbabcacbb=aabbaccbcacbcabc