⟨a, b | abaaba=aabaa⟩

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Monoids with two generators and one relation

#5844 ⟨a, b | abaaba=aaabb⟩

#5846 ⟨a, b | abaaba=aabab⟩

Properties

Presentation has sum-of-sides 11

Infinite non-cancellative non-commutative monoid

Not left cancellative, because left multiplication by a is not injective:

a ⋅ ba²ba = a²ba² and a ⋅ aba² = a²ba², however ba²ba ≠ aba²

Not right cancellative, because right multiplication by a is not injective:

aba²b ⋅ a = a²ba² and a²ba ⋅ a = a²ba², however aba²b ≠ a²ba

Enveloping group: ⟨a, b | aaabb⟩

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 = aabaa
Reduction order:
- Left-to-right recursive path with deg(c) = deg(b) = 0, c < b; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	cbc⁴ ⇒ c⁴bc	[15]
2.	c(bc)³ ⇒ c⁴	[13]
3.	abc⁴ ⇒ ac³bc	[14]
4.	a(bc)³ ⇒ ac³	[11]
5.	ca ⇒ abc	[6]
6.	cba ⇒ ac	[7]
7.	abac² ⇒ a²(bc)²	[9]
8.	aba²bc ⇒ a²bac	[16]
9.	aba²c³ ⇒ a²bac(bc)²	[17]
10.	a⁴c³ ⇒ (cbc)²	[18]
11.	a⁴(bc)² ⇒ cbc²	[12]
12.	a³bac ⇒ cbc	[8]
13.	a²ba² ⇒ c	[2]
14.	aba³c³ ⇒ c³bc	[19]
15.	aba³(bc)² ⇒ c³	[10]
16.	(aba)² ⇒ c	[3]

# ab:abaaba=aabaa cb/a aabaa=c magic:0 cbcccc=ccccbc cbcbcbc=cccc abcccc=acccbc abcbcbc=accc ca=abc cba=ac abacc=aabcbc abaabc=aabac abaaccc=aabacbcbc aaaaccc=cbccbc aaaabcbc=cbcc aaabac=cbc aabaa=c abaaaccc=cccbc abaaabcbc=ccc abaaba=c

#5845 ⟨a, b | abaaba=aabaa⟩

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Properties

Rewriting system

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