⟨a, b, c | aba=b, accb=1⟩

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Monoid enumeration

#2608 ⟨a, b, c | aba=b, acca=1⟩

#2610 ⟨a, b, c | aba=b, accc=1⟩

Properties

Sum of relation sides is 8

Infinite non-cancellative non-commutative monoid

Element a has infinite order

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

c ⋅ cda³c = c and c ⋅ 1 = c, however cda³c ≠ 1

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

ac²a ⋅ d = c²d and c² ⋅ d = c²d, however ac²a ≠ c²

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

#	Rule	Proof
1.	ada ⇒ d	[7]
2.	ad² ⇒ d²a	[8]
3.	ac²d ⇒ c²da	[23]
4.	a³d ⇒ dc²d	[13]
5.	c²d² ⇒ da³	[18]
6.	dc²da ⇒ a²d	[12]
7.	ac²ad ⇒ c²d	[22]
8.	dc²ad ⇒ adc²d	[20]
9.	c²dad ⇒ da²	[16]
10.	c²da³ ⇒ 1	[19]
11.	ac²a²d ⇒ c²ad	[21]
12.	dc²a²d ⇒ a²dc²d	[15]
13.	c²da²d ⇒ da	[10]
14.	b ⇒ da²	[5]

# abc:aba=b,accb=1 reversed:cad/b aab=d morph:3/0 ada=d add=dda accd=ccda aaad=dccd ccdd=daaa dccda=aad accad=ccd dccad=adccd ccdad=daa ccdaaa=1 accaad=ccad dccaad=aadccd ccdaad=da b=daa

#2609 ⟨a, b, c | aba=b, accb=1⟩

Contents

Properties

Rewriting system

Up:	Monoid enumeration
Prev:	#2608 ⟨a, b, c \| aba=b, acca=1⟩
Next:	#2610 ⟨a, b, c \| aba=b, accc=1⟩