⟨a, b | ababab=baaba⟩

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Monoids with two generators and one relation

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

#	Rule	Proof
1.	ce² ⇒ dc	[15]
2.	ec² ⇒ d	[7]
3.	ecdc ⇒ de²	[16]
4.	dc³ ⇒ ced	[17]
5.	dc²dc ⇒ cede²	[22]
6.	de⁴ ⇒ ecd²c	[24]
7.	e²a ⇒ c²ae	[21]
8.	dca ⇒ e³	[19]
9.	c³a ⇒ e²	[14]
10.	ceda ⇒ de²	[25]
11.	e²ca ⇒ c²ac³	[29]
12.	dc²a ⇒ edc	[20]
13.	dcea ⇒ cdae	[26]
14.	deda ⇒ ecde²	[23]
15.	dceca ⇒ cdac³	[34]
16.	de³da ⇒ ecd²e²	[27]
17.	ca² ⇒ ae	[8]
18.	(ca)² ⇒ ac³	[12]
19.	da² ⇒ ecae	[10]
20.	daca ⇒ ecac³	[28]
21.	ba² ⇒ e	[4]
22.	baca ⇒ c³	[11]
23.	eb ⇒ bac	[9]
24.	dcb ⇒ c⁵	[18]
25.	c³b ⇒ bac²	[13]
26.	cedb ⇒ dbac²	[31]
27.	dc²b ⇒ ecbac²	[30]
28.	dedb ⇒ ecdbac²	[32]
29.	de²cb ⇒ (ec)²bac²	[35]
30.	de³db ⇒ ecd²bac²	[33]
31.	ab ⇒ c	[2]

# ab:ababab=baaba ced/a/b ab=c,baacc=d,baa=e morph:2/0,5/1,3/0 cee=dc ecc=d ecdc=dee dccc=ced dccdc=cedee deeee=ecddc eea=ccae dca=eee ccca=ee ceda=dee eeca=ccaccc dcca=edc dcea=cdae deda=ecdee dceca=cdaccc deeeda=ecddee caa=ae caca=accc daa=ecae daca=ecaccc baa=e baca=ccc eb=bac dcb=ccccc cccb=bacc cedb=dbacc dccb=ecbacc dedb=ecdbacc deecb=ececbacc deeedb=ecddbacc ab=c

Other submonoids of same group

Σ	#	Presentation	Description
7	210	⟨a, b \| aaabb=ba⟩	Infinite cancellative non-commutative monoid
8	434	⟨a, b \| aaabab=ba⟩	Infinite cancellative non-commutative monoid
8	583	⟨a, b \| baaa=abab⟩	Infinite cancellative non-commutative monoid
9	903	⟨a, b \| aaabaab=ba⟩	Infinite cancellative non-commutative monoid
9	1120	⟨a, b \| ababab=baa⟩	Infinite cancellative non-commutative monoid
10	1883	⟨a, b \| aaabaaab=ba⟩	Infinite cancellative non-commutative monoid
11	3957	⟨a, b \| aaabaaaab=ba⟩	Infinite cancellative non-commutative monoid
11	4787	⟨a, b \| abaabaab=baa⟩	Infinite cancellative non-commutative monoid

Presentation

Description

210

⟨a, b | aaabb=ba⟩

Infinite cancellative non-commutative monoid

434

⟨a, b | aaabab=ba⟩

Infinite cancellative non-commutative monoid

583

⟨a, b | baaa=abab⟩

Infinite cancellative non-commutative monoid

903

⟨a, b | aaabaab=ba⟩

Infinite cancellative non-commutative monoid

1120

⟨a, b | ababab=baa⟩

Infinite cancellative non-commutative monoid

1883

⟨a, b | aaabaaab=ba⟩

Infinite cancellative non-commutative monoid

3957

⟨a, b | aaabaaaab=ba⟩

Infinite cancellative non-commutative monoid

4787

⟨a, b | abaabaab=baa⟩

Infinite cancellative non-commutative monoid

Other submonoids of anti-isomorphic group

Σ	#	Presentation	Description
7	245	⟨a, b \| aaab=bba⟩	Infinite cancellative non-commutative monoid
9	942	⟨a, b \| aababab=ba⟩	Infinite cancellative non-commutative monoid
9	1247	⟨a, b \| abaab=baaa⟩	Infinite cancellative non-commutative monoid
10	2571	⟨a, b \| abaaab=baaa⟩	Infinite cancellative non-commutative monoid
11	4086	⟨a, b \| aabaabaab=ba⟩	Infinite cancellative non-commutative monoid
11	4265	⟨a, b \| abaababab=ba⟩	Infinite cancellative non-commutative monoid
11	5289	⟨a, b \| abaaaab=baaa⟩	Infinite cancellative non-commutative monoid

Presentation

Description

245

⟨a, b | aaab=bba⟩

Infinite cancellative non-commutative monoid

942

⟨a, b | aababab=ba⟩

Infinite cancellative non-commutative monoid

1247

⟨a, b | abaab=baaa⟩

Infinite cancellative non-commutative monoid

2571