#23309 ⟨a, b | aaa=1, babb=abab⟩

Properties
Rewriting system

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 ⋅ ab⁴ = bab and b ⋅ ab = bab, however ab⁴ ≠ ab
Not right cancellative, because right multiplication by b is not injective:
- aba ⋅ b = bab² and bab ⋅ b = bab², however aba ≠ bab

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 shortlex with b < a
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a³ ⇒ 1	[1]
2.	(ab)² ⇒ bab²	[2]
3.	bab⁴ ⇒ bab	[3]
4.	(bab)² ⇒ (ab²)²	[4]
5.	b⁵ab ⇒ b²ab	[13]
6.	ab⁴ab ⇒ b⁴ab²	[11]
7.	bab³ab ⇒ a(ab²)²	[5]
8.	ba²b³ab ⇒ a²b³ab²	[14]
9.	b²a²b²ab ⇒ (ab³)²	[8]
10.	aba²b²ab ⇒ ba(ab²)²	[15]

# ab:aaa=1,babb=abab reversed:ba - -
aaa=1
abab=babb
babbbb=bab
babbab=abbabb
bbbbbab=bbab
abbbbab=bbbbabb
babbbab=aabbabb
baabbbab=aabbbabb
bbaabbab=abbbabbb
abaabbab=baabbabb

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#23309 ⟨a, b | aaa=1, babb=abab⟩

Contents

Properties

Rewriting system