⟨a, b | abab=aaa, bbbb=1⟩

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Monoid enumeration

#10942 ⟨a, b | aabb=bba, bbbb=1⟩

#11016 ⟨a, b | abba=aaa, bbbb=1⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element (a²b)²ba has infinite order

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

a ⋅ bab = a³ and a ⋅ a² = a³, however bab ≠ a²

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

a¹⁸ ⋅ a = a⁷ and a⁶ ⋅ a = a⁷, however a¹⁸ ≠ a⁶

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(a) = 0; deg(b) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	a¹⁹ ⇒ a⁷	[17]
2.	a¹⁷ba ⇒ a⁵ba	[16]
3.	a¹⁶ba² ⇒ a⁴ba²	[14]
4.	aba³ ⇒ a⁴b	[3]
5.	(ab)² ⇒ a³	[1]
6.	a¹⁶b²a ⇒ a⁴b²a	[13]
7.	(aba)² ⇒ a⁶b²	[7]
8.	a⁴b²a² ⇒ a¹⁴ba²b	[18]
9.	a³b³ ⇒ aba	[4]
10.	a⁴b²ab ⇒ a⁴ba²	[5]
11.	a⁴ba²b²a ⇒ a⁹ba²b²	[10]
12.	b⁴ ⇒ 1	[2]
13.	a⁴ba²b³ ⇒ a⁴b²a	[8]

# ab:abab=aaa,bbbb=1 reversed:a/b - - aaaaaaaaaaaaaaaaaaa=aaaaaaa aaaaaaaaaaaaaaaaaba=aaaaaba aaaaaaaaaaaaaaaabaa=aaaabaa abaaa=aaaab abab=aaa aaaaaaaaaaaaaaaabba=aaaabba abaaba=aaaaaabb aaaabbaa=aaaaaaaaaaaaaabaab aaabbb=aba aaaabbab=aaaabaa aaaabaabba=aaaaaaaaabaabb bbbb=1 aaaabaabbb=aaaabba

#10963 ⟨a, b | abab=aaa, bbbb=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#10942 ⟨a, b \| aabb=bba, bbbb=1⟩
Next:	#11016 ⟨a, b \| abba=aaa, bbbb=1⟩