⟨a, b | aaa=1, baba=abbb⟩

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Monoid enumeration

#23300 ⟨a, b | aaa=1, baab=abbb⟩

#23309 ⟨a, b | aaa=1, babb=abab⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element aba has infinite order

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

b ⋅ (ab²)² = b⁷ and b ⋅ b⁶ = b⁷, however (ab²)² ≠ b⁶

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

(bab)² ⋅ b = b⁷ and b⁶ ⋅ b = b⁷, however (bab)² ≠ b⁶

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

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

#	Rule	Proof
1.	b¹⁰ ⇒ b⁷	[39]
2.	ab⁷ ⇒ b⁸	[40]
3.	bab⁴ ⇒ b⁹	[37]
4.	b⁵ab² ⇒ b⁸	[51]
5.	b⁶ab ⇒ b⁸	[52]
6.	b⁷a ⇒ b⁸	[42]
7.	ba²b³ ⇒ ab⁴a	[4]
8.	(ba)² ⇒ ab³	[2]
9.	b(ab²)² ⇒ b⁷	[47]
10.	bab³ab ⇒ b⁷	[50]
11.	(b²a)²b ⇒ b⁷	[43]
12.	b⁵a²b ⇒ b⁸	[45]
13.	a³ ⇒ 1	[1]
14.	a(ab)² ⇒ b³a²	[5]
15.	ab³a² ⇒ bab	[3]
16.	ab⁶a² ⇒ (bab)²	[6]
17.	b²a²b²ab ⇒ b⁸	[48]
18.	b⁴ab²a²b ⇒ b⁷	[53]
19.	ab⁴a²ba ⇒ ba²b²ab³	[10]
20.	b(b²a²)² ⇒ a²b²ab	[8]
21.	b⁴(a²b)² ⇒ b⁷	[54]
22.	(a²b)²bab³ ⇒ b⁴a²ba	[19]
23.	(ba²)²b²ab ⇒ b⁷	[44]

# ab:aaa=1,baba=abbb b/a - - bbbbbbbbbb=bbbbbbb abbbbbbb=bbbbbbbb babbbb=bbbbbbbbb bbbbbabb=bbbbbbbb bbbbbbab=bbbbbbbb bbbbbbba=bbbbbbbb baabbb=abbbba baba=abbb babbabb=bbbbbbb babbbab=bbbbbbb bbabbab=bbbbbbb bbbbbaab=bbbbbbbb aaa=1 aabab=bbbaa abbbaa=bab abbbbbbaa=babbab bbaabbab=bbbbbbbb bbbbabbaab=bbbbbbb abbbbaaba=baabbabbb bbbaabbaa=aabbab bbbbaabaab=bbbbbbb aabaabbabbb=bbbbaaba baabaabbab=bbbbbbb

#23303 ⟨a, b | aaa=1, baba=abbb⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#23300 ⟨a, b \| aaa=1, baab=abbb⟩
Next:	#23309 ⟨a, b \| aaa=1, babb=abab⟩