#27655 ⟨a, b | aa=1, ababbb=bab⟩

Properties
Rewriting system
Isomorphic instances

Properties

Sum of relation sides is 11
Infinite non-cancellative non-commutative monoid
Element b has infinite order
Not left cancellative, because left multiplication by b is not injective:
- b ⋅ ab⁵ = bab and b ⋅ ab = bab, however ab⁵ ≠ ab
Not right cancellative, because right multiplication by b is not injective:
- bab⁴ ⋅ b = bab and ba ⋅ b = bab, however bab⁴ ≠ 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.	bab⁵ ⇒ bab	[4]
2.	b⁶ab ⇒ b²ab	[14]
3.	a² ⇒ 1	[1]
4.	(ab)² ⇒ bab³	[3]
5.	ab⁵ab ⇒ b⁵ab³	[16]
6.	bab³ab ⇒ ab²ab³	[5]
7.	b²ab⁴ab ⇒ ab⁴ab³	[9]
8.	b(b²a)²b ⇒ (ab³)²	[11]
9.	a(b²a)²b ⇒ (b²a)²b³	[12]

# ab:aa=1,ababbb=bab b/a - -
babbbbb=bab
bbbbbbab=bbab
aa=1
abab=babbb
abbbbbab=bbbbbabbb
babbbab=abbabbb
bbabbbbab=abbbbabbb
bbbabbab=abbbabbb
abbabbab=bbabbabbb

Isomorphic instances

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

1 total

Σ	#	Presentation	Mapping
11	28265	⟨a, b \| aa=1, babbb=abab⟩	φ(a) = a, φ(b) = b

Up:	Monoid enumeration
Prev:	#27649 ⟨a, b \| aa=1, ababba=bbb⟩
Next:	#27657 ⟨a, b \| aa=1, ababbb=bbb⟩

#27655 ⟨a, b | aa=1, ababbb=bab⟩

Contents

Properties

Rewriting system

Isomorphic instances