⟨a, b | aa=1, abbabba=bb⟩

Up:

Monoid enumeration

#27141 ⟨a, b | aa=1, abbabab=bb⟩

#27156 ⟨a, b | aa=1, abbbabb=bb⟩

Properties

Sum of relation sides is 11

Infinite non-cancellative non-commutative monoid

Element ca has infinite order

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

b ⋅ b⁵ = b² and b ⋅ b = b², however b⁵ ≠ b

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

b⁵ ⋅ b = b² and b ⋅ b = b², however b⁵ ≠ 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) = 0; deg(c) = deg(a) = 1, c < a
Certificate: derivations of all rewriting rules from the defining relations.
Morphocompletion: how the auxiliary generators were found.

#	Rule	Proof
1.	b⁶ ⇒ b²	[22]
2.	cb⁵ ⇒ cb	[24]
3.	bcb⁴ ⇒ bc	[23]
4.	b⁴c ⇒ cb⁴	[27]
5.	ab² ⇒ b³cb	[18]
6.	bab ⇒ c	[3]
7.	b²a ⇒ bcb³	[19]
8.	c² ⇒ b²cb²	[17]
9.	cb²c ⇒ b²	[29]
10.	cba ⇒ b²cb	[13]
11.	bca ⇒ b(cb³)²	[28]
12.	acb ⇒ (b³c)²b	[26]
13.	abc ⇒ bcb²	[14]
14.	bac ⇒ cab	[5]
15.	a² ⇒ 1	[1]

# ab:aa=1,abbabba=bb b/ca bab=c morph:3/0 bbbbbb=bb cbbbbb=cb bcbbbb=bc bbbbc=cbbbb abb=bbbcb bab=c bba=bcbbb cc=bbcbb cbbc=bb cba=bbcb bca=bcbbbcbbb acb=bbbcbbbcb abc=bcbb bac=cab aa=1

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

Σ	#	Presentation	Mapping
11	27672	⟨a, b \| aa=1, abbabb=bba⟩	φ(a) = a, φ(b) = b
11	28281	⟨a, b \| aa=1, bbabb=abba⟩	φ(a) = a, φ(b) = b

Presentation

Mapping

27672

⟨a, b | aa=1, abbabb=bba⟩

φ(a) = a, φ(b) = b

28281

⟨a, b | aa=1, bbabb=abba⟩

φ(a) = a, φ(b) = b

#27144 ⟨a, b | aa=1, abbabba=bb⟩

Contents

Properties

Rewriting system

Isomorphic instances

Up:	Monoid enumeration
Prev:	#27141 ⟨a, b \| aa=1, abbabab=bb⟩
Next:	#27156 ⟨a, b \| aa=1, abbbabb=bb⟩