⟨a, b | aa=1, abbabb=bb⟩

Up:

Monoid enumeration

#10008 ⟨a, b | aa=1, ababbb=bb⟩

#10020 ⟨a, b | aa=1, abbbab=bb⟩

Properties

Sum of relation sides is 10

Infinite non-cancellative non-commutative monoid

Element ac has infinite order

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

b ⋅ b³ = b² and b ⋅ b = b², however b³ ≠ b

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

b³ ⋅ b = b² and b ⋅ b = b², however b³ ≠ b

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

#	Rule	Proof
1.	c²b ⇒ b³	[7]
2.	c³ ⇒ b²c	[8]
3.	b⁴ ⇒ b²	[17]
4.	b³c ⇒ bc	[16]
5.	b²cb ⇒ cb	[11]
6.	b²c² ⇒ c²	[14]
7.	cb³ ⇒ cb	[18]
8.	cb²c ⇒ c²	[19]
9.	ab² ⇒ bcb	[6]
10.	abc ⇒ bc²	[12]
11.	acb ⇒ b(cb)²	[13]
12.	ac² ⇒ bcbc²	[15]
13.	bab ⇒ c	[3]
14.	cab ⇒ bac	[5]
15.	a² ⇒ 1	[1]

# ab:aa=1,abbabb=bb bc/a bab=c morph:3/0 ccb=bbb ccc=bbc bbbb=bb bbbc=bc bbcb=cb bbcc=cc cbbb=cb cbbc=cc abb=bcb abc=bcc acb=bcbcb acc=bcbcc bab=c cab=bac aa=1

The mapping is from the listed presentation's alphabet to the current rewriting system's alphabet.

Σ	#	Presentation	Mapping
10	10335	⟨a, b \| aa=1, bbabb=abb⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

10335

⟨a, b | aa=1, bbabb=abb⟩

φ(a) = a, φ(b) = b

#10016 ⟨a, b | aa=1, abbabb=bb⟩

Contents

Properties

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#10008 ⟨a, b \| aa=1, ababbb=bb⟩
Next:	#10020 ⟨a, b \| aa=1, abbbab=bb⟩