#22284 ⟨a, b | aaa=1, abbbab=bb⟩

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 ⋅ b²⁵ = b² and b ⋅ b = b², however b²⁵ ≠ b
Not right cancellative, because right multiplication by b is not injective:
- ab³a ⋅ b = b² and b ⋅ b = b², however ab³a ≠ b

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.	b²⁶ ⇒ b²	[14]
2.	b²⁴ab² ⇒ ab²	[13]
3.	ab⁵ ⇒ b⁴ab	[4]
4.	ab²ab ⇒ b²¹ab²	[16]
5.	ab³ab ⇒ b²	[2]
6.	ab⁴ab ⇒ b³ab⁴	[6]
7.	a²b² ⇒ b³ab	[3]
8.	abab² ⇒ b⁷(ab)²	[8]
9.	a³ ⇒ 1	[1]
10.	b²(ab)³ ⇒ b¹²ab³	[15]

# ab:aaa=1,abbbab=bb reversed:b/a - -
bbbbbbbbbbbbbbbbbbbbbbbbbb=bb
bbbbbbbbbbbbbbbbbbbbbbbbabb=abb
abbbbb=bbbbab
abbab=bbbbbbbbbbbbbbbbbbbbbabb
abbbab=bb
abbbbab=bbbabbbb
aabb=bbbab
ababb=bbbbbbbabab
aaa=1
bbababab=bbbbbbbbbbbbabbb

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#22284 ⟨a, b | aaa=1, abbbab=bb⟩

Contents

Properties

Rewriting system