⟨a, b | aaba=a, bbabb=a⟩

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Monoid enumeration

#14589 ⟨a, b | aaba=a, bbaba=a⟩

#14593 ⟨a, b | aaba=a, bbbaa=a⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

a ⋅ b = ad and a ⋅ d = ad, however b ≠ d

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

ada ⋅ d = d and 1 ⋅ d = d, however ada ≠ 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

Auxiliary generators:
1. c = bb
2. d = abab
Reduction order:
- Right-to-left recursive path with deg(d) = 0; deg(a) = deg(c) = deg(b) = 1, a < c < b
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	ad² ⇒ dad	[11]
2.	cd ⇒ dc	[23]
3.	dbd ⇒ dc	[21]
4.	(ad)² ⇒ d	[12]
5.	ab ⇒ ad	[9]
6.	adb ⇒ ac	[13]
7.	cb ⇒ bc	[6]
8.	b² ⇒ c	[3]
9.	a²da ⇒ a	[15]
10.	cada ⇒ db	[20]
11.	dbada ⇒ db	[22]
12.	a²c ⇒ ca²	[7]
13.	adac ⇒ db	[10]
14.	a²dc ⇒ ca²d	[14]
15.	cac ⇒ a	[5]
16.	cadc ⇒ ad	[17]
17.	dbac ⇒ ada²	[18]
18.	dbadc ⇒ ada²d	[19]

# ab:aaba=a,bbabb=a reversed:d/acb bb=c,abab=d morph:2/0,4/0 add=dad cd=dc dbd=dc adad=d ab=ad adb=ac cb=bc bb=c aada=a cada=db dbada=db aac=caa adac=db aadc=caad cac=a cadc=ad dbac=adaa dbadc=adaad

#14591 ⟨a, b | aaba=a, bbabb=a⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#14589 ⟨a, b \| aaba=a, bbaba=a⟩
Next:	#14593 ⟨a, b \| aaba=a, bbbaa=a⟩