⟨a, b | aaa=1, bbbb=abba⟩

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Monoid enumeration

#23325 ⟨a, b | aaa=1, bbbb=abab⟩

#23327 ⟨a, b | aaa=1, bbbb=abbb⟩

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 c is not injective:

c ⋅ b = c⁶ and c ⋅ c⁵ = c⁶, however b ≠ c⁵

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

b ⋅ c = c⁶ and c⁵ ⋅ c = c⁶, however b ≠ c⁵

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

Auxiliary generators:
1. c = bbbbb
Reduction order:
- Left-to-right recursive path with deg(c) = 0; deg(b) = 1; deg(a) = 2
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	c¹³ ⇒ c	[45]
2.	bc ⇒ c⁶	[50]
3.	cb ⇒ c⁶	[54]
4.	b² ⇒ c¹⁰	[52]
5.	cac¹² ⇒ ca	[41]
6.	cab ⇒ cac⁵	[53]
7.	c⁶a ⇒ ac⁶	[27]
8.	bac ⇒ c⁵ac	[51]
9.	a²c ⇒ c⁴ac¹¹	[47]
10.	ac²a ⇒ c²ac⁸	[40]
11.	ac⁴a ⇒ c²	[22]
12.	ca² ⇒ c⁵ac¹⁰	[48]
13.	a³ ⇒ 1	[1]

# ab:aaa=1,bbbb=abba c/b/a bbbbb=c morph:5/0 ccccccccccccc=c bc=cccccc cb=cccccc bb=cccccccccc cacccccccccccc=ca cab=caccccc cccccca=acccccc bac=cccccac aac=ccccaccccccccccc acca=ccacccccccc acccca=cc caa=cccccacccccccccc aaa=1

#23326 ⟨a, b | aaa=1, bbbb=abba⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#23325 ⟨a, b \| aaa=1, bbbb=abab⟩
Next:	#23327 ⟨a, b \| aaa=1, bbbb=abbb⟩