⟨a, b | aaa=a, babb=aba⟩

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Monoid enumeration

#19835 ⟨a, b | aaa=a, babb=aab⟩

#19837 ⟨a, b | aaa=a, babb=abb⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element b²a² has infinite order

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

a ⋅ a² = a and a ⋅ 1 = a, however a² ≠ 1

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

a² ⋅ a = a and 1 ⋅ a = a, however a² ≠ 1

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

#	Rule	Proof
1.	bab⁶ ⇒ bab²	[3]
2.	aba ⇒ bab²	[2]
3.	(ab²)² ⇒ bab³a	[5]
4.	bab³ab⁴ ⇒ bab³a	[7]
5.	(bab⁴)² ⇒ bab⁵a	[9]
6.	a³ ⇒ a	[1]
7.	bab²a² ⇒ bab²	[4]
8.	bab⁴a² ⇒ bab⁴	[6]
9.	b(b³a)³ ⇒ a(b³a)²	[32]
10.	(b⁵a)²b³a ⇒ bab⁵ab³a	[27]
11.	b(b⁴a)²b⁵a ⇒ bab⁴ab⁵a	[26]
12.	bab³a²b² ⇒ a(b³a)²	[8]
13.	(b⁵a)²ab² ⇒ bab⁵a²b²	[28]
14.	a(b³ab)²b ⇒ (b³a)³	[35]
15.	bab⁵a²b⁶ ⇒ bab⁵a²b²	[12]
16.	a²(b³a)² ⇒ bab⁵a²b²	[10]
17.	a(b³a)³ ⇒ bab⁵a²b³a	[39]
18.	ab(b²ab³)²a ⇒ (b⁴a)²b⁵a	[31]
19.	ab⁴ab⁵ab³a ⇒ b³ab⁵a²b²	[41]
20.	b(ab⁵a)² ⇒ (b³a)³	[25]
21.	(ab⁴)³ba ⇒ b(b²ab³)²a	[33]
22.	ab³ab⁵a²b² ⇒ b⁴ab⁵ab³a	[34]
23.	ab⁴ab⁵a²b² ⇒ b³ab⁵a²b³a	[36]
24.	bab⁵a²b⁴ab² ⇒ bab⁴ab⁵a	[40]
25.	bab⁵a²b³ab⁴ ⇒ bab⁵a²b³a	[15]
26.	bab³(b²a²)² ⇒ bab⁵a²b²	[13]
27.	bab(b⁴a²)² ⇒ bab⁵a²b⁴	[14]
28.	bab⁴(ba²b²)² ⇒ b(b²ab³)²a	[16]

# ab:aaa=a,babb=aba reversed:b/a - - babbbbbb=babb aba=babb abbabb=babbba babbbabbbb=babbba babbbbbabbbb=babbbbba aaa=a babbaa=babb babbbbaa=babbbb bbbbabbbabbba=abbbabbba bbbbbabbbbbabbba=babbbbbabbba bbbbbabbbbabbbbba=babbbbabbbbba babbbaabb=abbbabbba bbbbbabbbbbaabb=babbbbbaabb abbbabbbbabb=bbbabbbabbba babbbbbaabbbbbb=babbbbbaabb aabbbabbba=babbbbbaabb abbbabbbabbba=babbbbbaabbba abbbabbbbbabbba=bbbbabbbbabbbbba abbbbabbbbbabbba=bbbabbbbbaabb babbbbbaabbbbba=bbbabbbabbba abbbbabbbbabbbbba=bbbabbbbbabbba abbbabbbbbaabb=bbbbabbbbbabbba abbbbabbbbbaabb=bbbabbbbbaabbba babbbbbaabbbbabb=babbbbabbbbba babbbbbaabbbabbbb=babbbbbaabbba babbbbbaabbaa=babbbbbaabb babbbbbaabbbbaa=babbbbbaabbbb babbbbbaabbbaabb=bbbabbbbbabbba

#19836 ⟨a, b | aaa=a, babb=aba⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
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