#23325 ⟨a, b | aaa=1, bbbb=abab⟩

Properties
Rewriting system

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element ba² has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ ab²⁷ = bab³ and b ⋅ ab³ = bab³, however ab²⁷ ≠ ab³
Not right cancellative, because right multiplication by b is not injective:
- a²b³ ⋅ b = bab and ba ⋅ b = bab, however a²b³ ≠ ba

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.	b³⁰ ⇒ b⁶	[12]
2.	bab²⁷ ⇒ bab³	[13]
3.	b²ab²⁶ ⇒ b²ab²	[11]
4.	b⁴ab ⇒ ab⁵	[4]
5.	a²b⁴ ⇒ bab	[3]
6.	(ab)² ⇒ b⁴	[2]
7.	(ab²)² ⇒ b³ab⁵	[5]
8.	(bab)² ⇒ ab⁸	[6]
9.	bab³ab ⇒ ab³ab⁸	[7]
10.	a³ ⇒ 1	[1]
11.	a²b³ab² ⇒ b²ab²³	[10]

# ab:aaa=1,bbbb=abab b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbbbbbb=bbbbbb
babbbbbbbbbbbbbbbbbbbbbbbbbbb=babbb
bbabbbbbbbbbbbbbbbbbbbbbbbbbb=bbabb
bbbbab=abbbbb
aabbbb=bab
abab=bbbb
abbabb=bbbabbbbb
babbab=abbbbbbbb
babbbab=abbbabbbbbbbb
aaa=1
aabbbabb=bbabbbbbbbbbbbbbbbbbbbbbbb

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#23325 ⟨a, b | aaa=1, bbbb=abab⟩

Contents

Properties

Rewriting system