⟨a, b | bab=aaa, bbbb=b⟩

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Monoid enumeration

#13095 ⟨a, b | bab=aaa, baab=b⟩

#13100 ⟨a, b | bab=aba, aaaa=a⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element a³b has infinite order

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

b ⋅ b³ = b and b ⋅ 1 = b, however b³ ≠ 1

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

b³ ⋅ b = b and 1 ⋅ b = b, however b³ ≠ 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:
- Left-to-right recursive path with deg(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a²⁹ ⇒ a⁵	[18]
2.	aba²⁸ ⇒ aba⁴	[16]
3.	a⁴b ⇒ ba⁴	[3]
4.	b²a⁵ ⇒ aba²⁶	[14]
5.	bab ⇒ a³	[1]
6.	ba²ba⁴ ⇒ a³ba⁷	[8]
7.	ab²a⁴ ⇒ a²ba²⁵	[15]
8.	aba³ba⁴ ⇒ ba³ba¹¹	[11]
9.	a³b²a³ ⇒ ba⁴	[6]
10.	b³a³ ⇒ a³	[5]
11.	b²a³ba⁴ ⇒ a²ba²⁵	[17]
12.	a³b³ ⇒ a³	[4]
13.	b⁴ ⇒ b	[2]

# ab:bab=aaa,bbbb=b a/b - - aaaaaaaaaaaaaaaaaaaaaaaaaaaaa=aaaaa abaaaaaaaaaaaaaaaaaaaaaaaaaaaa=abaaaa aaaab=baaaa bbaaaaa=abaaaaaaaaaaaaaaaaaaaaaaaaaa bab=aaa baabaaaa=aaabaaaaaaa abbaaaa=aabaaaaaaaaaaaaaaaaaaaaaaaaa abaaabaaaa=baaabaaaaaaaaaaa aaabbaaa=baaaa bbbaaa=aaa bbaaabaaaa=aabaaaaaaaaaaaaaaaaaaaaaaaaa aaabbb=aaa bbbb=b

#13099 ⟨a, b | bab=aaa, bbbb=b⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#13095 ⟨a, b \| bab=aaa, baab=b⟩
Next:	#13100 ⟨a, b \| bab=aba, aaaa=a⟩