⟨a, b | aba=bb, aaaaa=a⟩

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Monoid enumeration

#15904 ⟨a, b | aba=ab, bbbbb=b⟩

#15908 ⟨a, b | aba=bb, aaaab=b⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element b²a² has infinite order

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

a ⋅ a⁴ = a and a ⋅ 1 = a, however a⁴ ≠ 1

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

a⁴ ⋅ a = a and 1 ⋅ a = a, however a⁴ ≠ 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(b) = 0; deg(a) = 1
Certificate: derivations of all rewriting rules from the defining relations.

#	Rule	Proof
1.	b¹⁶ ⇒ b⁴	[16]
2.	bab¹⁵ ⇒ bab³	[18]
3.	b³a ⇒ ab³	[3]
4.	aba ⇒ b²	[1]
5.	ba²b¹⁵ ⇒ ba²b³	[19]
6.	bab²ab³ ⇒ a²b⁸	[9]
7.	b²a²b³ ⇒ ab²ab¹³	[15]
8.	a³b⁴ ⇒ bab¹⁴	[17]
9.	ba³b³ ⇒ b²ab¹³	[14]
10.	ba²b²ab³ ⇒ a²b²ab⁸	[12]
11.	b²a³b² ⇒ ab³	[6]
12.	a⁴b² ⇒ b²	[5]
13.	a³b²ab³ ⇒ ba²b³	[8]
14.	b²a⁴ ⇒ b²	[4]
15.	a⁵ ⇒ a	[2]

# ab:aba=bb,aaaaa=a b/a - - bbbbbbbbbbbbbbbb=bbbb babbbbbbbbbbbbbbb=babbb bbba=abbb aba=bb baabbbbbbbbbbbbbbb=baabbb babbabbb=aabbbbbbbb bbaabbb=abbabbbbbbbbbbbbb aaabbbb=babbbbbbbbbbbbbb baaabbb=bbabbbbbbbbbbbbb baabbabbb=aabbabbbbbbbb bbaaabb=abbb aaaabb=bb aaabbabbb=baabbb bbaaaa=bb aaaaa=a

#15905 ⟨a, b | aba=bb, aaaaa=a⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#15904 ⟨a, b \| aba=ab, bbbbb=b⟩
Next:	#15908 ⟨a, b \| aba=bb, aaaab=b⟩